Some of what I want out of Final Fantasy:

1) Ability to customize characters. This means equipment for example. Preferably head/body/shield/weapon/accessory slots. This does not mean a grid of circles where you pick a circle when you level up. Finding equipment upgrades in treasure chests is really fun. Seeing a big list of stuff you want to buy in a store is really fun, it gives you a goal to work for.
2) World Map with a boat or airship I get to drive around on it.
3) A story that make sense.
4) Memorable, distinct characters that I give a crap about.

FF13 went off the deep end with made-up words and crap. I couldn't follow the plot at all. Some people got cursed and now they're wandering around. Some people I never met got kidnapped and we're trying to save them. I don't know who the villain of the game was. There were big monsters controlling people, there was some guy's giant head who showed up and shot lasers out of his eyes every once in a while[3], but I don't remember who he was. It was a freaking mess. Where was FF13's Golbez/Kefka/Sephiroth/Edea?

In FF13 you have a 3-minute movie [1] of that doo-rag wearing guy flying around in a space ship with his girlfriend and they pretty much scream "I LUUUUUUUUUUURRRRRRRVEEEEE YOUUUUUUUU" and start making out while J-Pop plays and fireworks and explosions go off in the background. I couldn't possibly care less. It's contrived and they're beating you upside the head with it.

Whereas I just played FF6 again last month, and Locke and Celes have an interesting relationship (given the small amount of dialog in the game) because it's subtle and has a little mystery to it. It's not the point of the game, the point of the game is to play the game. You become attached to the characters because you have so much fun fighting battles with them. You customize them and grow them and they become yours. I don't need to see Locke and Celes flying around on a spaceship.

I looked up the script for FF6; do you know how many lines of dialog Shadow has in that game? 38. And many of them are "Go!" or "Run!". But you remember his character because the character (like the game) has atmosphere. Same for most of the other characters.

Surely everyone has seen the Plinkett Star Wars review [2] where he talks about how no one in the Phantom Menace actually has a personality. Do the same test on FF13.

FF6:
Terra: Half-esper girl who doesn't know where she belongs, is trying to learn how to form relationships with people
Locke: Thief who wants to protect people because of his girlfriend being sick or whatever
Celes: Ex-badguy, morally conflicted, independent and strong
Kefka: Insane psycho clown

FF7:
Cloud: Self-doubting guy with issues who works through them to become strong eventually
Barret: Loud, pushy macho guy with a caring side for his daughter/friends
Cid: Foul-mouthed guy with leadership skills and a tragic past that left him angry, or something
Sephiroth: Is Sephiroth.

FF13:
Lightning: Admittedly badass. Also... grumpy?
Vanille: Dunno.
Sazh: I remember he had a chocobo on his head and had guns.
Fang: No freaking clue. I forgot she was in the game until I just looked her up. Tall and had an accent. Was friends (sister? girlfriend?) with Vanille.
Snow: I know he had a girlfriend and was voice-acted by Troy Baker.
Villain: A giant head with laser eyes.

Maybe it's because I haven't played FF13 as much as those others, but all the characters seemed so generic and forgettable.

The best Final Fantasy of this generation was probably Lost Odyssey. The characters were silly-looking, but very memorable. The character customization was great. The dream sequences were impressive plot and character development for a video game. The world had interesting places to explore; it was pretty much riding the rails JRPG style, but there were cool environments. You got a boat and you got to drive it around! Battles were just turn-based with a timed-hit system, but it was fun enough and challenging at times. The villain was ridiculous, but at least the game had a clear villain with a clearly-defined goal and the PCs had good personal motivation to be doing things to stop him. The game had simple, well-executed parts that fit together well.

[1]: Final Fantasy XIII - Serah and Snow Fireworks Scene [Japanese] [HD] - YouTube
[2]: Star Wars: The Phantom Menace Review (Part 1 of 7) - YouTube
[3]: Final Fantasy XIII Chapter 9 Boss Fight 2 Barthandelus - YouTube