One example is The Chain. I prefer the Three Days Grace cover over Fleetwood Mac's original....despite the fact that the original should be iconic to any Formula One fan living in the UK ! I also prefer Soft Cell's cover of Tainted Love over the Gloria Jones original (though I don't really care for Marilyn Manson's cover).
Do you mean usually it's only when you're actually playing the game, or did you mean that you usually only listen to the music from games that you play?
Some of the Crusader Kings II music is very good. There's only 90 minutes of music, and I'm not bored of it yet, despite having 200+ hours on the game!