The funny thing about my answer to this question is that the post-apocalyptic game I've enjoyed most isn't actually post-apocalyptic.
I'm talking, of course, about Rifts. The game certainly is (or at least was) presented as post-apocalyptic in its original form, but even back in the core rulebook, if you really look at it, the game's pretty far removed from conventional elements that typically characterize the post-apocalyptic genre. Dungeonskull Mountain does a great job of summarizing the points of why Rifts isn't actually post-apocalyptic, even though even Palladium has made extensive use of such imagery.
|That bus is looking pretty good for having sat out in the sun for 300 years...|
I think I enjoy the genre (whether cod-PA like Rifts or "purist" like AW) because it's almost like a subset of the horror genre; the sense that this is where we could very well be heading, either via a bang like nuclear war or a whimper like post-oil/post-scarcity collapse. There's a sense of possible realness to the genre that lends a frisson of recognition to the proceedings, but leavened somewhat by its alien reality (very much so, in the case of Rifts) and even moments of slight comedy - who can resist the temptation of throwing in degenerate cults that worship Elvis or having pre-cataclysm hordes stuffed full of cheap junk that people now regard as priceless?