I actually bought the Japanese version once it came out, but I was occupied by other games at the time and didn't get to it. When the American version was released I also bought that, but again waited for several months before picking it up. Once I did I must admit I've had difficulties putting it down. As a genre Dissidia is essentially a fighting game, and I think it fulfills the requirements quite nicely. Supposedly the mechanics were based off of Advent Children -- i.e. flying all over the place during battle -- and I have to say it works pretty well. I actually find it to be a difficult game (I still haven't beaten Inward Chaos), and I swear to god the computer pulls shenanigans: It's capable of moves which are physically impossible for the in-game mechanics. Chaos himself probably has a dozen moves which are impossible to block or dodge. It's hair-rippingly annoying at times, but fun. The only level which pisses me off is Final Fantasy II's simply because the corridors sometimes creates terrible camera angles. I can't tell you how many times I've been in the midst of a chase and I can't see shit.
Dissidia is a crossover game, and as a general rule crossover games' plots are crap. It's impossible to put mash together characters from different universes and have it make a coherent plot. I remember as I played Radiant Mythology, it interesting to see how Namco tried to explain mana, fonons, lens, and force abilities in the same place (they did it by emphasizing mana and ignoring the other systems mostly). Even for a crossover game I feel Dissidia is worse than most in terms of plot. There's barely any substance to it, and the dialogue tries to come off as deep but ends up as sounding pretentious. I'm actually okay with this because, as I said, essentially it's a fighting game, and we all know they're known for great plots.
Sometimes they even revamped some of the personalities of the characters. I would say RPGs didn't really take off as a genre until the SNES days because earlier systems didn't have the memory to make a coherent plot. As systems become better there's more space to develop story and characters. Square had to create personalities for the characters of their earlier games just for Dissidia because they simply didn't have one. Hell, the Warrior of Light is a Dissidia-only character since Final Fantasy I didn't have a strict protagonist. I do find Butz to be particularly odd though because it seems they completely changed him. I don't remember him to be cowardly at all, but everyone acts as if that's what he does all the time.
In terms of graphics it's a rather pretty game (as usual of Square). Particularly interesting is looking at the earlier characters because, let's face it, you couldn't see much in those little sprites. I really dislike Amano Yoshitaka's art style (that's for another day), but it's still better than what you usually experience.
Also interesting is if you put your PSP into French, the American version will change its language to French as well. This doesn't work for any language.
All in all, I have to give it a good review. Something I'll credit Square with is in recent games they've tried to work with different battle systems, and this one was surprisingly enjoyable and interesting. Good work.