So I've been re-reading the Harry Potter series and finished my favorite from that franchise: Harry Potter and the Prisoner of Azkaban. Unlike its predecessors it doesn't try to be a fairy tale; unlike its successors, it doesn't try to be a sprawling epic. It's the only book whose villain isn't some incarnation of Voldemort, and it sports a unique dual-climax: first we get the reveal of who Sirius Black really is, then we get the time-traveling adventure that results in Wormtail's escape and Black's freedom.
The book introduces the dementors, who will remain the most fearsome creatures in the series. We also get to meet Remus Lupin who, along with Black, give us a window into Harry's past while being amongst the most complex and well-rounded characters in the series. Both Black and Lupin are tired, tormented crusaders, trapped by their idealism, misunderstood by their society, and hostage to their need to defend their loved ones. Additionally, Lupin's relationship with Snape hints at a conflict that won't be fully understood until the final few chapters of Book 7. We also get a prophecy from Professor Trelawny, which Dumbledore says is the second real one she's ever made. We learn about the first in Book 5.
The film version of Azkaban also stands out for me, although in re-watching it I'm less enamored with it than before. It's substantially better than the first two, but I think I may like Half-Blood Prince better. Director Alfonso Cuarón focused on making the main characters real and placing them in a fantastical world, although the fantasy of that world registers as severe strangeness in some peripheral characters, like the driver of the Knight Bus (or his talking shrunken head).
There are some significant breaks from the novel: Cuarón stripped away absolutely everything that wasn't necessary to the plot, truncated quite a few scenes and filled the rest with character moments. The Quidditch subplot is done away with almost entirely. In the book, it's a major affair, taking up several chapters and ultimately resulting in a win for Potter. This was important in the scope of the series: Harry was supposed to this great Quidditch player, but his team always ended up losing the House Cup. They had pretty much gone as far as they could without a major victory to justify Harry's greatness. In the movie, it's reduced to two brief scenes that are plot-essential: the dementors torment Harry, and the whomping willow destroys his broom.
The way dementors and petronus' are dealt with differs greatly from the book. The dementors in the movie fly, and the emotional "coldness" Harry feels when they are near is translated into literal coldness on screen. Things freeze when they're around. This was, of course, a cinematic necessity (I don't know about the "flying" bit, but whatever). But the most significant and impressive changes deal with the way Cuarón wrangled the double-climax ending.
Notably, he gave Harry more to do: in the book it's Lupin who discovers that Peter Pettigrew still lives, but in the movie Harry makes that discovery and then tells Lupin. During the confrontation with Black, the movie leaves the audience guessing for a while about who's good and who's bad. That scene in the book is much more straight-forward, with Lupin constantly asking Harry to let him explain. During the time-travel sequence, Harry and Hermione interact more with themselves--saving their own lives, as it were. This allows the first climax to pass less dramatically, setting up mysteries to be revealed once Harry and Hermione go back into the past. It gives them more to do when they're replaying the evening (in the book, they basically hide in the woods and watch the action happen--it works, but it would be dreadfully boring to watch in a film). This keeps the tension high and the pace brusk, preventing what could have become a very dull, protracted ending while not sacrificing major plot elements.
My only real complaint about the movie is that a few scenes aren't terribly well acted. The leads were still too young to pull of some of the more complex scenes believably.