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.

I probably should have mentioned "You Got What it Takes (to Take What I Got)" by Joe Tex, later covered as a duet by Marti Jones and her husband, Don Dixon. My wife and I used the latter version as our first dance.

The Feelies used to end their live shows with great cover songs - Everybody's Got Something to Hide but Me and My Monkey (Beatles), King's Lead Hat (Brian Eno), All Tomorrow's Parties (Velvet Underground). Etc.

A few stealth covers from Cyndi Lauper - "Money Changes Everything", covering The Brains, and "All Through the Night" her cover of Jules Shear.

Strange variation 1:  "World Turning", Lindsey Buckingham and Christine McVie's re-work of Peter Green's "World Keep on Turnin".  almost unrecognizable and uncredited to Green.

Strange Variation 2 -  "Battleship Chains" Georgia Satellites (Terry Anderson) covering The Woods (Terry Anderson's first band) and later covered by Hindu Love Gods (basically, REM plus Warren Zevon.)

Special Mention: Kid Creole's irresistibly danceable yet hilarious cover version of Harry Belafonte's "If You Wanna Be Happy For the Rest of Your Life (Never Make a Pretty Woman Your Wife).

Slaw's mention of Alejandro Escovedo brings another cool one to mind - his cover of Mott the Hoople's "All The Young Dude's".  I'm not sure that he's ever recorded it, but he sometimes ends his live show with an audience sing along version of the song that is terrific fun.