Most cover tunes are just an unknown band's way of getting noticed. The following, however, actually added something new through their remakes:
5. Mad World - Michael Andrews
This cover took a B-side from the group Tears for Fears, which was originally recorded as an upbeat pop song, and made it into a haunting ballad (appeared on soundtrack of the movie Donnie Darko).
4. Hazy Shade of Winter - The Bangles
Okay, I admit it. I had a crush on Susannah Hoffs back in the 80s. So sue me. This cover, also appearing in a movie (Less Than Zero) took a little known Simon and Garfunkel song and made their own. Great harmonies from an underrated group of vocalists.
3. Dancing in the Dark - Pete Yorn
Appears as a bonus track on Music for the Morning After, this cover of Bruce Springstein's 80's hit changes the tempo in a manner that is really more in line with the lyrics than the original version.
2. Land of Confusion - Disturbed
When Genesis recorded this song, they had really moved from rock band to pop band. Heck, the video for the original featured the "Spitting Image" puppets. Disturbed took the song and made it rock, and their video looks like something out of Pink Floyd's The Wall.
1. Wild Horses - The Sundays
If you have not heard this cover, do yourself a favor - find the video on YouTube and give it a listen. This version puts the Rolling Stones original - itself a classic - to shame. One of the best vocal performances you'll ever hear.
Any others I've left out?

