Song covers that are better than the originals

Shonen Knife: Top of the World
Nazareth: Love Hurts (and I love Orbison)
Persuasions: Angel of Harlem
Cowboy Junkies: Thunder Road
Eagles: Hotel California (tell me this is not a steal/cover of Jethro Tull's "We Used to Know")

How about Jazz "covers" of popular tunes, such as:

Coltrane "My Favorite Things"

Johnny Griffin "The Way You Look Tonight"

One more for the list:

The Ballad of Lucy Jordan by Marianne Faithfull. I like that version much more than the overwrought version by Dr. Hook.

Yet one more to add to the list:

Odetta covering Dylan's "Times they are a-Changin"

I like Marianne Faithfull's "Sister Morphine" more than the Stone's version, but, I don't know which is the cover (Jagger and Faithfull co-wrote the song).

Peter Paul and Mary's "Leaving on a Jet Plane" is another decent cover.

I like covers that change the tempo and nature of the song radically. Though not better than the originals, I still like My Chemical Romance's uptempo "Desolation Row" and Nouvelle Vague's down tempo "Guns of Brixton."  For weird and fun, anything covered by Shonen Knife will be fun to hear once in a while.