One great genre that uses this is burgeoning field of Urban Fantasy. Buffy the Vampire Slayer. World of Darkness. Jim Butcher's Dresden Files. Most of the SciFi series knocking around. The world is just like it is outside, but with succubi and werewolves. And actually, it is a little more comforting if you know your garbage cans are being raided by werewolves than by those pesky raccoons.
Similarly, Spy games use the real world, often with unintended results and comparisons. TSR's Top Secret had an adventure, Operation: Lady in Distress, which featured the hijacking of a luxury cruise ship. It used the plans for the same ship type as the Achille Lauro, which was hijacked two years later.
|The place seems familiar, put I can't put a name on it.|
Then there is the Near Future, with the example of Shadowrun, where the power of magic returning to the world would create a Seattle filled with shamans, trolls, cyber-enhanced netrunners, and elves with slugthrowers. In the original, the magic returned to the world in 2011, which was three years ago, which is the danger of putting your future to near to your pub date. The game also described a group of terrorists blowing up the Sears Tower, which sounded a lot more science-fictiony before 9/11.
The Past, of course, is another country, and it is here that Call of Cthulhu thrives. The "core" campaign is set in the 20s, with offshoots in the modern age, the 1890s, and various other times (there has been a lot of products in the more depressing 30s than the ironic, world-is-about-to-crash 20s. I am particularly fond of the 20s, as the adventures are as much history lesson as they are Mythos hunting. As a result, I usually peruse the internet, trying to determine such minutia as how to place a telephone call and when automatic starters started showing up in cars, replacing the old hand crank.
The advantage of all of these campaigns is that there is a wealth of information. The disadvantage is that, well, there is a wealth of information. People actually KNOW how things works in a real world, such that while in a traditional fantasy campaign you might not call the local guard, in an Urban Fantasy campaign you would probably call the cops if there were sudden explosions in the areas (unless, of course, the cops were really WEREWOLVES).
So consider this a broad catagory of possible campaigns, where there are subtle and not-so-subtle changes, and secret layers operating beneath the surface of the world. And that world would be the one outside your window.