I pretty much agree with Gamblet and Sephex. Plot twists, characterization, and overall story are what I like about FF. Nothing really stands out in particular since the FF stories aren't that original. But they are fun.

One thing I do like is the whimsical nature of some of the earlier games. Characters like Ultros and Gilgamesh really break up the 'Grrr must save the world super srs' nature of story that RPGs typically have. I feel that's really missing from the later games.