Despite Cuarón's popularity, he denies rumors that he'll return to the Harry Potter franchise. In 2014, he revealed that he would not direct the Potter spin-off films based on Fantastic Beasts and Where to Find Them. Instead, David Yates, director of many other Harry Potter films, will tell us where to find them. (Source)

There's something queer about Lupin's little speech after he resigns. And by "queer" we mean both "strange" and "not heterosexual." The reason for the parallels between Lupin's speech and that of a homosexual teacher being forced to resign after being outed is because director Alfonso Cuarón read the character as gay. He instructed actor David Thewlis to play the role as a "gay junkie" jonesing for the moon. (Source)

Author M. Joseph Young's fascinated with time travel, and he analyzed the heck out of the time travel portion in the finale of Prisoner of Azkaban. Our eyes crossed about halfway through, but Young's ultimate verdict is that the time travel, as presented in the novel, doesn't work. But, "the filmmakers blundered into a solution to the anomaly that works." Good job, guys. (Source)

They couldn't make all the Harry Potter books into two films, so many elements from the book must be left out to squeeze the plot into one two-hour movie. This site lists all the differences between the book and movie. And because the film makers were able to tell the future (the book series was up to Order of the Phoenix by the time this film was made) they were able to add hints to Ron and Hermione's budding relationship, which wasn't in the book. (Source)