Final Fantasy IX is my favorite Final Fantasy (and quite possibly my favorite game) by a large margin. As far as I'm concerned, it has the best soundtrack in the series, bar none. Granted, a lot of the songs are just different versions of the same tune, but there are many others that are absolutely excellent.
I cared more about the characters in FFIX more than I've ever cared about a character in a game. I felt that a majority of the characters had more depth than others in previous games. The only other game I believe to have come close in character development is Final Fantasy VI. All the characters in FFIX played a significant role in the plot. I never really felt as if anyone got left behind (as I believed happened in FFVII and FFVIII). As FFVII was the first Final Fantasy I played (I didn't start getting my hands on the older ones until after I played FFIX), I was incredibly relieved to have a main character who wasn't a stereotypical loner.
The animation style never bothered me, really. The characters would have certainly looked a lot cooler had they stuck more closely to Amano's concept art (Steiner looks like a total bad ass), but I believe the animation fits the attitudes of the characters more.
One of the best parts of the game, I believe, is the villain. Kuja is simply awesome. He's the only villain in a Final Fantasy game I've ever pitied. I felt he had real motivation, and it was refreshing to see a villain who wasn't trying to become a god to enslave the world, but instead someone who was acting on real emotions.
I loved the reintroduction of the class system. I've always preferred distinct definition between characters and their roles in battle. Choosing your party becomes less of a "These guys are my favorite, so I'll just use them," and instead introduces at least some element of strategy.