Part one: Characterization via gameplay.

In Final Fantasy IV, each character's gameplay traits match what the story wants us to think about them. Though present in other games, no other Final Fantasy in the NES, SNES, or PS1 era does this so well. Final Fantasy IV has an internally consistent set of gameplay rules and mechanics, and each character's unique story traits are adapted to this system.

(spoilers follow)

Cecil has committed evil deeds and feels guilt, he begins as a Dark Knight weilding evil weapons. He and Kain, experienced warriors, start at level 10 and face foes that are much lower level. Their abilities are direct and powerful, presenting only a few options in each battle. The perfect way to begin the game.

Rydia, the child-summoner, starts out at level 1, knowing only one summon spell and has enough MP to cast it just one time. By the end of the game, well... you all know how she ends up. Though she will usually have the lowest HP of anyone in the group.

Tellah, an aging Wizard with extensive experience and the power of both white and black magic, whose skills have deteriorated. He has an arbitrary, fixed selection of spells that aren't learned in any sort of progression, and changes only once during a gameplay event. His fixed MP and low ability scores (due to age) prevent his extensive spell selection from being overpowered early in the game. Gives you a preview of what is to come with Rydia and Rosa.

Edward is weak and a coward who runs away in battle and while his bard skills are useful, they're hardly powerful. For once, a game that includes a unique bard character class without any pretense that it's potential matches that of, say, a Paladin or Summoner.

Rosa is pure and good, and is a white wizard. Her archery ability is a minor anomaly, not really explained well by the story.

Yang, a straightforward monk character, a high-ranking one from a monk-centric kingdom. He's stoic, loyal, and very strong. He's differentiated from fighters by stats and reduced weapon/armor selection.

Palom & Porom, twins with a fun and powerful gimmick ability. They are child prodigies who learn spells at an accelerated rate (relative, in particular, to Rydia and Rosa). Note how Palom & Porom are differentiated from the other wizards in the game without trying to give them a whole new set of abilities. They use the same equipment and cast the same spells as all the other wizard types.

Cid is primarily an engineer, in battle he is a straightforward fighter with a natural ability to identify enemy weak points. While it may seem uninteresting, a character like this provides a great contrast to the magic-using characters and also facilitates tailored encounter design, which I'll get to later. For now, suffice it to say that these simple battle skills fit his character just fine. The game would not have been any better had Cid been a tool-weilding character like Edgar from FFVI.

Edge, the Ninja character. Cocky and ambitious, his skills are considerable, but he's not nearly as strong as he thinks. Low HP, low defense, Rubicant defeats him easily. He basically replaces Yang in the well-rounded 5 character party, being a more interesting class to play but also more advanced, requiring a little more thought. But again I'm getting ahead of myself. Like Yang, he represents one of the game's particular cultures.

FuSoYa, the Lunarian. He's an old guy from the moon, a great excuse to give him every spell in the game. Not sure there's a really good story-based reason why his HP and MP are fixed but it's a minor point.

Of course, there are many points during the game where the story and supposed rules of the game world get a bit muddy: why does Tellah die and not Yang? But Final Fantasy narratives always bend their own rules a little, it's not worth fussing about.

Other games in the series don't do this as well.

In V and VII, this is a very deliberate design decision. V in particular makes very little pretense whatsoever that character's story traits correspond to gameplay traits. Crystals grant jobs, that's it. That way, they're free to make that job system as addictive and engaging as possible. And it certainly is addictive and engaging. In V, the characters are blank slates that you develop over the course of the game. The system in VII is worth a whole post on its own, suffice to say that character differentiation is present, but minimal.

VI is a weird one. On the one hand, character story traits are directly reflected in gameplay. Sabin is a martial artist and has martial arts skills. Cyan is a samurai and has samurai skills. Characters are further differentiated by base statistics (strength, magic power, etc.). On the other hand, most of the primary characters are all very similar in a few key ways. Everyone has similar HP and MP. Everyone can learn every spell. Character class abilities can be remarkably similar in practice (Cyan's dispatch vs Sabin's pummel), and aren't as tightly balanced against the enemes as is the case in FFIV.

I haven't played VIII, but my impression is that it's slightly more differentiated than VII but not by much. FF IX is the only other SNES and PS1 era game that does classes like FFIV.