Gender bender

+1 on the Sundays cover of Wild Horses. 

Also, Macy Gray's cover of Metallica's Nothing Else Matters is pretty good. 

Leonard’s Cohen’s "Alexandra Leaving" done solo by his long-time collaborator Sharon Robinson

This one is a close call but Holly Cole’s "I Want You" is about dead even with Tom Waits’ original.

Not sure if any would agree is truly "better" but certainly a whole different spin and genre change here.

Judas Priest cover of Joan Baez Diamonds and Rust.
Appeared on their Sin after Sin album.
Personally I love what they did with it.

Another toss up but...
"Sudden Stop"
by Percy Sledge or Lou Ann Barton