Music headphones will usually give clearer and more accurate sound reproduction. Gaming headphones will usually pay more attention to looks and (sometimes) positioning. I prefer the music phones myself, and Sennheiser is a great brand to consider (I have the EH350).
If you can go with some decent headphones you'll not look back and you won't even want cheap phones any more, they just won't cut it. Once you get a taste of what accuracy from a decent set of "cans" sounds and feels like (and even almost tastes like) you'll thank yourself. Good sound is "delicious". I learned that when I was 17...now I'm 38. Good sound is much more attainable from a set of headphones than it is from speakers, from a cost perspective, IMO.
As I said, the Sennheiser EH350 is a great set of budget headphones that still sound MUCH better than some of the junk out there. The HD555 or 595 headphones are great too - they should have a bit more bass than the 350s. Grado is nice too (I actually like their sound slightly better than the Sennheisers), but I've never found their funny design to be all that comfortable.