If you entirely skip over Dissidia and buy Duodecim, you're not missing anything at all. Duodecim is like an extended Dissidia with more story and characters and the assist system. As noted previously, Dissidia's story is also playable in Duodecim. So no worries, you got the right one.
Today I found a copy of both and I bought Duodecim. Am I missing much by not having bought Dissidia?
Well in the first one, the God of Discord (a.k.a Chaos) dies and you probably won't figure out how, though it is pretty obvious, and the Goddess of Harmony (a.k.a Cosmos) diss appears because of her warriors...so you're missing that and the song Chaos: Last Battle 3 (The original song is called "The Messenger" by Your Favorite Enemies, but in the game, Square ENIX decided to split it into 3 parts. The last part was the best!) You're also going to miss fighting the first Chaos, which was fun and hard. On Duodicium, he's so easy it's lame.stare
Duodecium is a prequel to Dissidia. And after you finish the twelfth cycle, you move to the thirteenth cycle (Dissidia's story).
So you technically bought two games in one. Dissidia is in Duodecium and can only be unlocked by beating the Duodecium first. The only difference between them is that you play with a more updated battle system that is featured in Duodeciuem.
Plus, more costumes. Shirtless Tidus became my favorite.