First and foremost, this is very much its own movie -- although it's spun from the same root, the fabric of this narrative is distinct from what you've seen in the animated version and/or stage production. (I have seen the stage version -- specifically, the first national touring production, which inherited some of its cast from the original Broadway run. I also own the Broadway cast album for the show, as well as a set of CDs including not just the animated soundtrack, but additional developmental material by Ashman and Menken.)
Think of it this way: the animated feature is essentially Belle's story -- she is the primary focus around whom everyone else orbits. It's the closest of the three to a pure romance, and the one of the three that really acknowledges the passage of time as the central relationship develops (compare "Something There" in each version, and you'll see what I mean). By contrast, in the stage version, it's Lumiere who's at the center of things -- it's his musical presence and status as nominal host that focuses the action, particularly in the second half of the show, driven in part by the expanded attention given to the Beast's houseful of transformed servants (notably in the long number surrounding "Human Again").
This shifts again in the live-action movie: this time, it's Gaston who's chief catalyst and motivator. It's Gaston, much more of a direct threat much earlier on than we've previously seen, who propels Belle and Maurice into the events that lead them to the Beast's castle. In this version, what happens to Maurice and to Belle is in significant part driven by Gaston's active antagonism, and likewise the Beast's actions are driven in part -- if less overtly -- by Gaston's influence. This is an altogether nastier, darker, more knowingly Evil Gaston than he's been before, and the whole movie is colored by that difference.
Which is not to say that all is dark and grim and moody, because it's not. "Be Our Guest" and the post-climax grand ball are still spectacular, the banter between Lumiere and Cogsworth is still delicious, and Kevin Kline injects some wonderfully dry wit into Maurice's character. Indeed, this is a more character-driven film than either of its prior counterparts -- we get more back story for all of the principals, including additional songs for Maurice and the Beast (in fact, very little music is borrowed from the stage musical). There is also some further context for the spell and its effects, with ramifications that are likely to generate a good deal of fanfic.
All in all, I liked the movie a great deal. Ironically, the one area where it doesn't quite sparkle is the singing, which just can't quite compete with the iconic performances from the animated original (although the live version of "Gaston" comes close). Don't get me wrong -- it's mostly very good, just not really at the level of either of its predecessors.