The essence of the post-apocalyptic genre is that civilization has lost in a big way. It's also a contrast between two pairs of ideas:
- freedom vs. community
- simplicity vs. technology
This means there has to be a pre-apocalyptic civilization, usually our own, that is a well-defined group with a pro-science attitude as a contrast. There must be a sense of exactly what is lost, so that you can wonder whether it was worth it, or if things could have been different. There's two approaches to this:
- early post-apocalyptic: several survivors of the apocalypse, usually including the main character, remember the Before Times.
- late post-apocalyptic: no one remembers the Before Times, but there's plenty of evidence left.
Post-apoc RPGs are usually of the "late" variety. Gamma World is built on the disconnect between character knowledge and player knowledge. Although Gamma World is meant to be taken seriously (as James suggests,) there's a lot of intentional humor derived from the players knowing about the pre-apocalyptic world. Hoops are not inherently funny; they're funny because we know they're really rabbits.
I can't think of many early post-apoc RPGs. If any exist, they'd probably seem grimmer than Gamma World; they'd play more like a horror RPG.
* The other Corman film I'm referring to is Teenage Caveman. It's presented as a story about cavemen frightened of a beast in the forbidden lands, but the end of the film reveals that these cavemen live in our future.