when trying to think of cover songs I like better than the originals, two immediately came to mind.
1) One More Cup of Coffee covered by The White Stripes (originally by Bob Dylan)
2) Careless Whisper covered by Gossip (originally by George Michael)
Me First And The Gimme Gimmes also crossed my mind, but I think it might be considered unfair/too easy to list a cover band here.
Tonight alive Little Lion Man way better then the Mumford and Sons version.
Heart of Glass covered by Lunachicks (originally by Blondie)
Opeth's version of Soldier of Fortune, hands down amazing.
More proof of when other metal bands cover a Metallica song from the Black Album, they tend to make it better (and it's not just Lemmy's cover of "Enter Sandman"). Here are some examples:
Finntroll - The God That Failed
ICS Vortex - Don't Tread On Me
Borknagar - My Friend Of Misery
Sodom - The Struggle Within
Doro - Nothing Else Matters