In the context of 4e D&D, as I experienced, classes serve as thematic "pathways to" or "bundles of" player-side engagement with the game.
The theme results from the mechanical abilities that flow from a particular class, and the associated fiction.
For example, being a cleric or a paladin means worshipping a god. And that manifest through abilities that tend towards the radiant - which is a reflection of the "fact" that, in the fiction of 4e D&D, gods mostly live or are connected to the Astral Sea and radiance is a manifestation of divine power. (The 4e DMG has a little section offering advice on how to change this feature of divine abilities to make it better fit clerics and paladins of deities, evil ones, who are not associated with the Astral Sea.)
Being a fighter means having a high armour class and this is generally reflected, in the fiction, by wearing heavy armour. (An alternative, but not a build I've personally seen in play, is to be a STR/DEX fighter in which case AC should still be good but in the fiction this would be understood as being quick on one's feet and adept at dodging attacks.)
In many ways I see this as similar to Gygax's AD&D, but with a few differences (which in my view are mostly strengths):
* Moreso than AD&D, there is a focus on player-side abilities that will support the thematic play at which the class is aimed. Eg in 4e, fighters have abilities that strongly encourage the player of a fighter to put his/her PC into the heart of any fray. Gygax expresses a similar view about how fighters should be played (PHB p 18: "Fighters generally seek to engage in hand-to-hand combat, for they have more hit points and better weaponry in general than do other classes; DMG p 86: "fighters who hang back from combat . . . or fail to boldly lead . . . are . . . clear examples of a POOR rating), but there is less in the actual design of class abilities - at least in the case of fighters and I think some other classes too - to help ensure that this will be an emergent consequence of playing the class well in a tactical/technical sense.
* Moreso than AD&D, these thematic aspects of classes are coupled to the default cosmology and campaign history. This is particularly visible in the warlock class for 4e, but even with fighters there is more of a sense in 4e of what it is about the fantasy world that brings it about that people like this exist in it. (A comparison could be made here with Monte Cook's Arcana Unearthed.)
* 4e is overall a bit more permissive than AD&D (especially for human characters) in allowing combinations of classes, either via multi-class rules or hybrid rules or both. As the PHB 3 expressly warns about hybrid characters, they generate a risk of creating an unplayable character. I think there is a corresponding risk of creating a thematically dislocated or incoherent character. But equally these can be used to create strong thematic characters, especially when the role of Paragon Paths and Epic Destinies is also considered. In my long-running game, for instance, we had two "paladins" (to use the term in a loose sense) though mechanically one of them was a paladin and the other was a fighter with a cleric multi-class feat and the cleric Warpriest Paragon Path. Both character were melee-oriented heavily armed and armoured god-fearing and divinely inspired warriors, ie paladins.