Before tackling that beast of a question, I want to point out I have not finished the game, I am about half way through (Ch 8ish), and am trying to avoid spoilers in the hopes that I like it better when I try playing it again.
The problems start with a lack of interaction with anything. Even NPCs would only kind of fade in and out of a conversation they were already having, you couldn't actually talk TO them.
The game just FEELS different with the map off VS. the map on. If you turn the map off, the game doesn't feel so much like you're in a tiny cage, but with that overlay defining everything that's on screen as pointless fluff, directing you to the next map, it makes it REALLY easy to see "through" the code and NOT get pulled in.
The battle system, which to me, hovers between tedious and ineffective, yet was touted as "the next big thing". Paradigm shifts suck. It'd be better if you could pick and choose what each Paradigm's abilities were, but they were more set in stone than Mt. Rushmore.
My biggest problem with it, however, was the writing/naming/acting. This girl is bland, we'll call her Vanilla, but change the last letter so nobody figures it out (the voice actress, it should be noted, can not pull off an Australian accent, but is actually FROM Australia, which I found confusing)! Falcie, False-E, Lecie, I still don't know what the hell those things actually ARE. They made a point to say the names over and over and over again like they're normal words, which is a trick to get people familiar with the place and setting, it's supposed to invoke mystery and awe, like "oh this is the kind of magical place where they use pseudo-french sounding words for no reason" when there are almost definitely plain english equivalents that would have made more sense and been easier to explain and therefor follow. Those are the summon esper things? Got it. The whatever whatever is the summon esper thing for this game. Fine. Stop saying it. I hate when sci fi type stuff uses fake words that a normal person can barely pronounce or spell.
Lightning, Snow, Fang, Hope. These ALL sound like horrible fanfiction copies of Cloud Strife. Two of them are WEATHER. But apparently Square Enix hired that horrible writer(s?) with a Noun Name obsession.
I didn't find the graphics to be all that great, honestly. The backgrounds are pretty, but so is my desktop wallpaper. The characters' faces and hair are still very polygonal and everyone looks like they're coated in oily plastic and hairspray. Uncanny valley to the MAX. The expressions don't really match the voices most of the time.
The music is acceptable, but nothing stands out. I like the story I think, but it jumps back and forth so much I don't really know what's going on.
Amongst all this, I find Sazh to be the most like-able, which is a shame, because he's clearly only there to be a token Will Smith black guy comic relief. Big monster machine thing comes up, we zoom in on Sazh to hear him say "Whooo?!" and run. It's not that much better than Barrett.
I don't hate the game, but one has to think about what exactly SquareEnix was thinking when they put it together.
All in all, I'd say the problems all stem from the lack of a unified vision among the developers of what the game should be, and the fact that all big decisions were obviously made by some sort of board or committee instead of someone who may have been able to make it all make sense.