I don't usually like listening to covers--they usually don't add much to the original song, but there are a few covers that I can say add a new dimension to the original or are even musically better.
Cover: Billie Jean - Chris Cornell
-Cornell adds a dimension of agony to this song which is present in the lyrics but not in the original performance/arrangement. I think that this cover adds a elucidates a dimension to this song that was already there and just waiting to be let out.
Original: Billie Jean - Michael Jackson
It's growing on me. Definitely more fun than the cover
Cover: Sweet Dreams - Marlyn Manson
The lyrics to this song are genuinely creepy, but the original music is errrr....80's pop style annoying.Marlyn fixes it.
Original: Sweet Dreams - Eurythmics
Not the biggest fan of this song. Creepy lyrics sung to a peppy little tune with little to no emotion.
Cover: Comfortably Numb - Scissor Sisters
This song takes the opposite approach than the above. It takes a pretty angsty rendition of a pretty angsty song and turns it into a really queer disco anthem. This one is good for tweaking noses, and it's a surprising and fun re-envisioning.
Original: Comfortably Numb - Pink Floyd
Themesong of angsty teens everywhere.
Cover: New Years Day - Frontline Assembly
FLA puts this song in a different key, "industrializes" it, and casts defunct 80's star Tiffany as the lead singer. The result is a wierdly familiar industrial song with stellar vocals and lots of surprises.
Original: New Years Day - U2
I like the original a lot. I suppose my only clue that it might need a cover was that it is wayyyy too hummable and predictable. Which is the mark of songs you can listen to obsessively for a few weeks and then never want to hear again.
I'm sure this list is going to grow as I think of more.