Same song different artist. Which ones do you love? LP

The first one that always comes to mind:

"Knockin' on Heaven's Door" by Dylan and the cover by Warren Zevon (done as he knew he was losing his battle with cancer).  The latter is almost painfully poignant.

I also have a weak spot for "Oops, I Did it Again" by Brittney Sears and the great (tho unlikely) cover version by Richard Thompson.

Off the top of my head...
"Take Me To The River" - Talking Heads
"Hard To Handle" - Grateful Dead
"Me and Bobby McGee" - Janis Joplin
"Just Like Heaven" - Katie Melua
"Where Is My Mind" - Trampled By Turtles
"Shenandoah" - Trampled By Turtles
"Take Me Home Country Roads" - Toots and the Maytals

Antje Duvekot's covers of "Mercedes Benz" & "Fat -Bottom Girls" (right up there w Martykl's nomination for most unlikely cover).
+1 for Zevon's cover of "Knockin' on Heavens Door". 
How about the many covers of Leonard Cohen's "Hallelujah", Dylan's "All Along the Watchtower", "Don't Think Twice It's Alright" & "I Shall Be Released", John Prine's "Angel from Montgomery", John Fogerty's "Lodi", Cyndi Lauper's "Time After Time".
Another favorite cover of mine is Emmy Lou Harris singing Stephan Foster's "Hard Times Come Again No More".

Van Morrison's cover of Comfortably Numb
+1 to Jimi Hendrix's version of All Along The Watchtower

Bettye Lavette / Most of the time 
Cat Power / Amazing Grace
Eddie From Ohio / I know you rider
Jennifer Warnes / Bird on a wire
Dave Matthews Band / All along the watchtower
Carter USM / Another brick in the wall
Les Claypool & Chris Robinson / Battle of New Orleans