Tori Amos - "Smells like teen spirit"
http://uk.youtube.com/watch?v=wcHNZVrxEts
Oysterband & June Tabor doing a gorgeous version of "Love will tear you apart".
http://uk.youtube.com/watch?v=YYPg8qCrNAY (rubbish sound quality, but you get the idea)
I'd say it's Jimi Hendrix' cover of The Star Spangled Banner at Woodstock. He "adapted" it in a way which was shocking back then, to symbolise the way America had screwed-up in relation to Vietnam
(where he had himself served), ie horror, shock, mayhem, drugs.
http://youtube.com/watch?v=MFLy8eGtSYo
Cowboy Junkies - Sweet Jane (Velvet Underground)
Jimi Hendrix - All Along the Watchtower (Bob Dylan)
The Byrds - It's All Over Now, Baby Blue (Bob Dylan)
Pearl Jam - Keep on Rockin' (Neil Young)
Roxy Music - Jealous Guy (Beatles - ok, go easy on this one)
And you know, we could probably list dozens of great covers. There are plenty of songs that are covered better than the originals. These two tried to stay true to the original and succeeded.