I have to say that "Prisoner of Azkaban" was my favorite because it did the greatest justice to the source material than any other film. Granted, there was that one bit at the end that occurred out of sequence, but I feel like it was integrated seamlessly into the story itself. Alfonso Cuaron did a magnificent job.
I remember reading an interview for the film, which was with Alfonso and Jo, and she herself said the scene at the Shrieking Shack blew her away, it gave her goosebumps. She couldn't say why, but now after all these years, we can understand why: The Shrieking Shack was such an important setting, particularly in Deathly Hallows. I found myself extremely disappointed that it was not incorporated into the Yates film.
While I know that this thread is to discuss which movie you liked best and why, I can't help but express my disdain for Yates's direction. OOTP, HPB, and even DH left a lot to be desired. I think the least harm came to Deathly Hallows, and even then I think that's because he had the adaptation split in two installments. Alas, I digress.
Prisoner of Azkaban remains, in my eyes, the best of the Harry Potter films and it is only with grief that I regard Alfonso's decision to not direct another film (if not all subsequent installments).