Just about anything by Dylan is better in the cover.

Phil Collins' cover of Diana Ross' "Can't Hurry Love."

And I'm not sure whether Kitten Sundae's cover of Kashmir is better than the Zep original, but it's pretty darn fine, and tends to threaten the rafters of whatever venue they're currently rocking.... Besides. Music followed by uncontrollable giggles? FTW!

(We really should come up with some shout-out for the gang just as they're about to launch into the thing, to incorrige them.... :) (spelling deliberate. They're incorrigible. :)

Marilyn Manson's cover of Soft Cell's version of "Tainted Love".

Or the Genitorture's cover of the Divinyl's cover of the Rolling Stone's "I Touch Myself"

REM's cover of Tommy James and the Shondells' "Draggin' the Line"

I would definitely second some of those, although I actually like Cohen's original version of "Everyone Knows" better than Concrete Blonde's.

Other covers I really like:

Coil's version of "Tainted Love"
Current 93's version of "Misery Farm"
Nightwish's version of "Over the Hills and Far Away"
Johnny Hollow's version of "People Are Strange"

This reminds me about the time when I compared and contrasted the cover songs done by "This Mortal Coil" with the original works by the original artists. I may be biased, but the "This Mortal Coil" covers are far better and more awakening than the originals. The artists doing the covers gave a depth and dimension to the works that is not seen in the original creations. Not to knock the originals, and the original artists mind you, their creations have merit and place as well.
I personally feel that a lot of music can benefit from changing hands and going from one artist/band to another. In a lot of respects the music evolves and gets better, creating something wonderful that nobody expects.
This does not work for all music and not all artist can take any given piece of music and make it work, let alone make it better. Covers are tricky like that.