Not too long ago, I came across an article on Stargazer's Blog: "Is it just me or is the post-apocalypse much like a fantasy world?" Since many of the stories featured here take place after the fall of human society, its probably a good idea to chime in on this. Its late in the day of course, so excuse the somewhat disjointed format of the essay.
Its a matter of loss, knowledge, and recovery.
There are certainly a lot of parallels between the two genres (Or if you want to take the Joesph Campbell approach - between all genres). Digging through ruins for items of great power. Strange creatures or bandits to be wary of, natural hazards to avoid, and most outings are for a quest rather than simply walking because its a nice day. (Would you go jogging every morning if there was a 15% chance of meeting gnolls?)
In a fantasy world, however, that has always been the case. Orcs have been around since time immemorial, dragons live centuries, and the goblin problem is a known issue. Either by design or just dint of our books focusing on the interesting parts - fantasy worlds are made for adventure.
We have worked long and hard to make sure the real world is not a place of adventure. Most of the human eating megafauna was killed off thousands of years ago, military might keeps raiders at bay.
Most PA settings do not take place centuries after the event. As such there are people who still remember the before times. Gunslingers understand that their weapons are machines and not magic - spare parts might be rare, but it can be fixed or replaced. Perhaps even improved upon. There is a sense that what is gone can be recovered.
Conclusively removing the fantasy elements from a fantasy setting is rarely the goal. Sometimes the quest is to actually stop the big bad evil guy from doing that. Meanwhile the goal of a PA setting is to make it less hostile to all life. Fantasy is protecting the status quo, PA is restoring it.