I think a mix of both can be useful, but always having, as you call them, "real songs" can get distracting. Also, both the music and the lyrics have to fit with what's going on in the game (in a general way), or it'll just be stupid. Usually I just use the default music, at least to start. In some of my newer games, I've been using actual songs for certain cutscenes, but I find having someone singing while you're just walking around is distracting to the gameplay.
And I realize that "actual songs" can still be instrumentals, but if I'm understanding your terms correctly, I don't think it'd be easy to find enough good instrumentals that aren't video game-y to fill a whole game.