I think maybe it was my buddy Stuart who once offhandedly referred to post-apocalyptic gaming as "fantasy in drag" or "D&D in drag" or something like that. Being the kind of guy I am, I immediately replied "that's why I like it". I meant what I said, but I thought I'd take a moment to consider exactly what I meant by that comment. Just so we're all on the same page, I should note that by "post-apocalyptic gaming" I mean primarily 1st and 2nd edition Gamma World, Jonathan Tweet's Omega World (a fab d20 reimagining of GW) and Mutant Future. Other games in the same neighborhood include Rifts, After the Bomb, World of Synnibarr, Encounter Critical and Low Life. Here's what I dig about them.
Wild, Wild West - The Western was one of the most popular film and TV genres back when Gygax and Arneson were kids and I've long considered the world of fictional cowboys and Indians to be one of the hidden influences on D&D. The theme of taming the wilderness comes up again and again. More important, though, is the general attitude towards civilization and law enforcement. The typical D&D world is a pretty violent and lawless place where PC's can be either the Solitary Marshal defending the weak or the desperate banditos laying waste to everything around them. Only once in a while do the cops arrive to screw everything up. Gamma World and its descendants usually take this approach as well, which I generally prefer to more civilized modern or sci-fi settings.
Random Chargen - Over the years I've grown to loathe most forms of character 'construction'. I'd generally rather play a twerp I diced up than an ubermensch I had to buy with points. Most post-apocalyptic games make character generation even more fun than D&D by means of that greatest of chargen devices, the random mutation table. I just love rolling some bones to find out what sort of gimpy wretch I'm playing tonight. Some players revel in showing up to the game with the perfectly constructed munchkin PC. I like taking a crappy one thrown together in five minutes and doing something with the cards Fate dealt to me. And sometimes you get to shoot gamma rays out of your eyeballs and totally fry people.
Lasers - I don't always groove on elves and lasers. But three-armed mutants and lasers go together like chocolate and peanut butter. I really don't know what else to say here. I just like lasers. Also, robots.
Heavy Handed Social Commentary Disguised As Stupid Jokes - You know what I'm talking about here. The GM hates football players so he builds a scenario around a ruined city where the inhabitants are ruled each year by one of two political factions, whichever one wins the Big Game. The leader of the winning team and new leader of the city is given the grand title Emveepee. Sometimes things like that are cute, sometimes they are lame. Either way I think such shenanigans work a lot better in a post-apocalyptic game than in a straight D&D set-up.