From what I've seen so far, its mostly just the story mode. There's also at least one new game mode in which you have a "party" and each member has "powers," which in story is only used for either "training" rounds or else for "boss fights."
I myself have not gotten far into the game and haven't explored everything so there may be more stuff. For me it was just nice to spend more time with Ringo and Co.
And sometimes you play modes that combine the two.
Yeah, seriously, there's two different modes that have you playing both at once: one is called Swap (you play one for about thirty seconds then swap to the other for thirty seconds, and they keep rotating until someone wins), the other is called Fusion.... that one takes some getting used to.
It's also possible to do mixed matches where one player is playing Tetris and the other is playing Puyo. I've heard this is unbalanced but, well, I wouldn't know.