Overall

First of all, even if it doesn't show in some parts of this text, I really enjoyed the movie!  It was good to see Hogwarts and all the characters alive again!  
However, I was very disappointed from the start when, the moment I was expecting Snape to drop on the boys because he had the intelligence to guess exactly what had happened to them, we saw no other than Filch! Maybe Snape sent Filch to get them, but it was never mentioned! (See my Book 2 analysis for more details)  Filch was a logical choice to replace Snape, but why did they have to do it in the first place? That event summarizes very well how the whole movie unfolded: everywhere Snape was supposed to do something important or show how cunning and intelligent he is, someone else did his job!  What's their problem?  No, really, this was so stupid!  Rickman could have done such a great job at it!  
 

Details
scene by scene

First scene - Discovering Potter and Weasley

I don't like that scene because it's so different!  The only thing I love from it is the facial expressions Rickman used to make Snape more tangible to the audience. Well, for once, Snape was not the one to catch the boys.  It was Filch!  That was so stupid, thank you.  Then, they got in front of Snape who read the journal.  At least, we can guess that he understood how they had come to Hogwarts!  But it was not as glorious as in the book!  Not to mention that one of my favourite quote wasn't there: So," he said softly, "the train isn't good enough for the famous Harry Potter and his faithful sidekick Weasley. Maybe it would have been too much like the book, but I'm sure they could have though of something fun for Snape to say!  Oh well!  Then, it was the same, Snape told them that he was sorry he was not their Head of House.  The way Rickman crawled on his desk to tell them that was very Slytherin like!  He was like a predator advancing on his preys.  That was great!  That's when Dumbledore and Minerva came in!  That's the only part where I felt Rickman was not emotional enough.  When he learned that the boys wouldn't be expelled, he didn't look like Christmas had been cancelled.  You could see he was fuming inside, but not in his face.  Not enough.  There wasn't enough venom in his eyes like it was mentioned in the book. But the rest was great!

Note about the new seeker

At least the note was not signed by Filch!  That's all I had to say about that!

Mrs. Norris

Here, Snape's part has not been altered so much! Wow! This is what was in the book: Potter and his friends may have simply been in the wrong place at the wrong time," he said, a slight sneer curling his mouth as though he doubted it. "But we do have a set of suspicious circumstances here. Why was he in the upstairs corridor at all? Why wasn't he at the Halloween feast?"  That's pretty much the same, except there was no Halloween involved. However, when Potter told him he had been serving detention with Lockhart, he looked ready to kill (thanks Leanne!).  I guess the stupidiest git on earth had just cancelled Snape's plans!! And at the same time, he looked disgusted that this excuse-for-a-man (who's replacing him by the way) should be doing such things! Oh boy!  I think Snape has noticed something, and since he believes Potter is synonym for trouble, well add 1 and 1 and you get 2! Snape didn't look furious as in the book, but still, Rickman managed to do a good job: he was self-controlled, had his cold and sarcastic look, he was prowling over his victims, and used his low silky voice, etc.

Polyjuice Potion

Strange how this dangerous potion from Most Potente Potions could be found so easily in the movie!  I know it would have taken too much time to steal it, but they could still have mentioned that it was Snape who had told them about that book some time ago!  Wouldn't have hurt, but it would have made Hermione look less resourceful for a second!

First Quidditch match

If I'm not mistaking, this is where the first Quidditch match occurs.  Snape is sitting next to Lucius Malfoy!  I think it's only normal he would.  I have not seen them talk or look at each other for that matter.  They were just there. One thing I loved about Lucius is when they showed his disgusted look when Draco fell from his broom.  I thought it was a nice shot from the producers.  That's one thing I enjoyed about the movie: they did cut Snape's parts, but they've put in a lot about the Malfoys.  These two seem more tri-dimensional now!

Swelling potion in Snape's classroom

It would have been fun to see the stealing of potions ingredients, but I guess they didn't have time to do it!  It's a part of the movie they just had to simplify!  Sadly, we didn't get a chance to see Snape's classroom again!  It would have been cool to see the swelling potion!  I wonder if they'll do that scene for the DVD?  That's a cool scene from every point of view, not only Snape fans!

The duel

Here is the best Snape part of the movie of course! It's the longest and Snape finally does something!  I'll be more lengthy there off.

I loved this scene of course!  Except when Snape's Serpensortia was stolen and given to Draco!

The duel between Lockhart and Snape is great!  We can feel the hate irradiating from Snape towards Lockhart just by the way he stands his ground, his cold attitude, not even flinching a bit, and his disinterested voice. I don't remember if Lockhart said something about not hurting Snape anymore!!  It was in the book, so I'm afraid I'm getting mixed up here. Anyway, the beginning of the duel was just perfect: Snape looked so serious, proud and ready to kick Lockhart's ass!  Then, he used his Expelliarmus spell on Lockhart!  (It was funny to see the look of Hermione then!)  Rickman looked fantastic in that stance! He took the same as would Malfoy and Potter afterwards.  It gave him an air of elegance and dignity.   Then of course, Lockhart did his speech about expecting to receive the Expelliamus spell all along!  There, Rickman remained as calm as possible, so knowing Snape, you knew he could have killed him on the spot.

Then happened something I really enjoyed: Rickman tried to look innocent when he told Lockhart not to use Weasley because, unless, they were going to send Potter to the infirmary in a matchbox!  Nice! Rickman tried to give himself an innocent look when he proposed "say Malfoy?"  I never expected that.  I thought that like in the book, Snape would be smiling instead of looking innocent.  But I love what Rickman did because it shows how Snape's trying to fool Lockhart! Snape just knows how to tackle Lockhart like a puppet on strings!  Here is what was said in the book. You'll see that the producers have only changed the names and circumstances to suit the movie: I think Id better teach you how to block unfriendly spells," said Lockhart, standing flustered in the midst of the hall. He glanced at Snape, whose black eyes glinted, and looked quickly away. "Let's have a volunteer pair - Longbottom and Finch-Fletchley, how about you -"

"A bad idea, Professor Lockhart," said Snape, gliding over like a large and malevolent bat. "Longbottom causes devastation with the simplest spells. We'll be sending what's left of Finch-Fletchley up to the hospital wing in a matchbox." Neville's round, pink face went pinker.

"How about Malfoy and Potter?" said Snape with a twisted smile. "Excellent idea!" said Lockhart, gesturing Harry and Malfoy into the middle of the hall as the crowd backed away to give them room.

So, Snape got at the end of the duelling table while Malfoy got on it.  From that moment on, Snape's not doing a lot, but he surely is expressing a lot!  First of all, when Draco and Potter take their battle stance, he looks at Harry, not at Malfoy.  I think he doesn't want to miss a single thing about Harry's butt being kicked up in the air!  Now, was Snape aware that Malfoy would try to hit Potter before the signal? I'm not sure.  That was obvious coming from Draco however, so maybe yes, Snape thought it possible. However, Snape didn't look offended about that move from his pupil. I don't think we saw his reaction to that because all the attention was focused on Harry.

Then, Harry hit Draco full force with a spell I can't remember!  That's when you see Draco flying in the air in Trailer B. That part was way cooler than in the book!  You could see Snape looking quite cold and distant from what had just happened to Malfoy. He didn't try to catch him at all. I think Rickman just tried to look abashed. Yet, something that was absolutely not part of the book happened: Snape grabbed Malfoy by his school robes and shoved him up in a matter of seconds so that Malfoy could retaliate. You could feel the disappointment in Snape that he should both have to pick up the boy and that he was hit full force by Potter. I think Snape knew that Malfoy, being the spoon fed baby he is, would stay longer on the floor than was needed. That scene was very Slytherin like: the mentor picks up his pupil in a rude manner because he's disappointed in him and also wants his pupil to know that he can't back off now. That was the right thing to do for anybody.  But what was Slytherin like about it was the grabbing of the clothes. A Gryffindor would have taken his pupil by the arm or hand. For a Slytherin, that would mean to show sympathy and comfort towards the pupil. That was not what both Draco and Snape wanted that's for sure.  That's why I think this part was great.

Then, Serpensortia occurred.  Snape looked quite proud of his pupil just then, didn't he?  I think we can understand the Slytherin philosophy just by that: trying isn't good enough. Succeed and you'll be awarded a smile. Slytherins are secretive after all!  

Then, the scene is brought to an end by Snape telling Potter that he will get rid of the snake for him.  You see the air of eloquence, pride and contentment radiating from Snape.  He's so happy that Harry doesn't know what to do with the snake.  The only problem in this scene is that the snake doesn't look like he's going to attack before Harry talks to it. That's my impression anyway. After, we get a couple of great shots of Snape looking quite taken aback by Harry's abilities as a parsel tongue!  Rickman's look could be described exactly as in the book: "Snape, too, was looking at Harry in an unexpected way: It was a shrewd and calculating look, and Harry didn't like it."

 Valentine Day

 Well, of course they didn't have time for that in the movie! It's a shame though!  They did get Lockhart other funny moments though!

After Dumbledore has been sent away by Lucius Malfoy

This scene does not appear at all because of time of course! This part was one of the few that gave Malfoy a little essence. But I think the producers did a good job at that even without this scene.  Here's an extract of it from the book:

I always thought Father might be the one who got rid of Dumbledore," he said, not troubling to keep his voice down. "I told you he thinks Dumbledore's the worst headmaster the school's ever had. Maybe we'll get a decent headmaster now. Someone who won't want the Chamber of Secrets closed. McGonagall won't last long, she's only filling in ......

Snape swept past Harry, making no comment about Hermione's empty seat and cauldron.

"Sir," said Malfoy loudly. "Sir, why don't you apply for the headmaster's job?"

"Now, now, Malfoy," said Snape, though he couldn't suppress a thinlipped smile. "Professor Dumbledore has only been suspended by the governors. I daresay he'll be back with us soon enough."

"Yeah, right," said Malfoy, smirking. "I expect you'd have Father's vote, sir, if you wanted to apply for the job - I'll tell Father you're the best teacher here, sir -"

Snape smirked as he swept off around the dungeon, fortunately not spotting Seamus Finnigan, who was pretending to vomit into his cauldron. 

"Her skeleton will lie in the Chamber forever"

Does he look like he has a plan or what!  

Believe it or not, they've given a bigger role to Snape compared to the other teachers in this scene!  Well, not that big, but still, he kept his place for once!!

I liked it because you saw that Snape was the main character and not Minerva even though she was there. Snape tipped off the other teachers so they would follow his plan of sending Lockhart away for good!  I think Snape knew Lockhart would flee in the face of danger. The only sad part of this scene, in both the movie and the book, is that we don't know what the other teachers will do.  I'm sure Snape will not leave this matter into the hands of Lockhart!  So, I think it's a shame we know nothing of what he tried to do to solve the mystery.  I bet he tried to look for the entrance himself, somewhere in the dungeons since Slytherins seem to always have been there!  

Here's what happened in the book.  I have coloured the parts which were attributed to Snape in the movie in red and what was said in blue. I'm not 100% sure though.
"So sorry - dozed off - what have I missed?" He didn't seem to notice that the other teachers were looking at him with something remarkably like hatred. Snape stepped forward.

"Just the man," he said. "The very man. A girl has been snatched by the monster, Lockhart. Taken into the Chamber of Secrets itself. Your moment has come at last."
Lockhart blanched.
"That's right, Gilderoy," chipped in Professor Sprout. "Weren't you saying just last night that you've known all along where the entrance to the Chamber of Secrets is?"
"I - well, I -"sputtered Lockhart.
"Yes, didn't you tell me you were sure you knew what was inside it?" piped up Professor Flitwick.
"D-did I? I don't recall -"
"I certainly remember you saying you were sorry you hadn't had a crack at the monster before Hagrid was arrested," said Snape. "Didn't you say that the whole affair had been bungled, and that you should have been given a free rein from the first?"

Expelliarmus

Again another place where Snape could have been mentioned but wasn't. When Harry used Expelliarmus on Lockhart in the book and told him : "Shouldn't have let Professor Snape teach us that one,"  

Mandrake Potion

In the book, the potion was to be brewed by no other than Snape, who had vehemently let know (to Lockhart) that no one else was to do it because he was the Potions Master.  But then, at the end, you hear Dumbledore ask a round of applause for Mrs. Sprout and Mrs. Pomphrey for brewing up the potion!  Gee! How much discredit to Snape did they needed in that movie?  

The End

At the end, where everybody applauses Hagrid, though they didn't explain why which was silly, I couldn't see if Snape was also applauding. I think not, but I'm not sure.  I think that maybe he did at first, but didn't go on as long as the other!  That would only be natural, wouldn't it?

Overall performance

Rickman was great, though it's hard to tell from the little screen-time he's had!  However, I think he did his best with that!  The first scene with him seems to me like he lacked the conviction, the joy of kicking Potter out and then the hate of not being able to do so. But due to all the changes that were made to this scene, I guess it's understandable! I couldn't see Snape fuming as I had expected he would, but Rickman did that with his face.  You could feel how Snape respects Dumbledore because he was not shouting at him at all or giving him orders. So, yes, Rickman did a great job with his facial expression!  Of course, each time we saw him, I was watching his every move.  I think Rickman understood that if he was not going to have any text to say, he was going to act it silently.  There were a lot of quick glances, glitters in the eyes, almost unperceivable movements with the head that gave Rickman a truly Snapish look.  I think that overall his performance as Snape could be described as such: cool, self-controlled, controlled sarcasm, expressionist, and cold.

Rumors about Rickman not wanting to participate in the other movies

Thanks to PIyanun, I've been informed that this was a rumor after all! I'm glad that's the only rumor I ever posted on my website, because that's the very reason why I hate them!  
Here is what I learned on the Leaky Cauldron website:
The Facer has posted a retraction of their post from earlier this week, that Alan Rickman was considering stepping away from future HP movies because he hadn't had enough to do. Thanks to Amanda from HPfGU and a few posts to various Rickman sites, so TLC could get confirmation, we refrained from publicizing the original statement, but now that it's been shown to be a fraud someone tried to perpetrate on TheFacer, we're happy to say that it's not true. Reuters reports that Rickman is expected to be at the premiere in London early next month.

Rickman's touch!

A thing that really caught my eye was how 2 or 3 times Rickman gave Snape a somewhat "innocent-look".  What I mean is that, when he used sarcasm like Snape always does, at the same time he tried to look innocent.  One such occasion was when Snape proposed Malfoy to duel against Potter.  It's like he tried to look stupid on purpose because he wanted Lockhart to agree to his plan!  I loved that little touch which I had not thought of when I read the book.