I've never heard a Beatles re-make that I preferred over the original. Ever.
I understand the preference for the original, but Jim Cocker's "With a Little Help From My Friends" was damn good. And I love Siouxsie & the Banshees' versions of "Dear Prudence" and the minimalist "Helter Skelter", but that's cuz I'm a huge Siouxsie fan.
Never Let Me Down Again - Depeche Mode (original) and Smashing Pumpkins(remake)
That's on my list.
Top rank has got to be Hendrix's cover of "All Along the Watchtower", right? What a classic.
Johnny Cash's cover of "Hurt" has to be up there too.
In turn, I enjoy Nine Inch Nails' cover of "Dead Souls" more than the Joy Division original. Same thing goes with Siouxsie & the Banshees' version of Iggy Pop's "The Passenger"; I love that cover.
Echo and the Bunnymen's "People Are Strange" sounds better than original, amazing as that may seem.
Guilty indulgence for me: Ciccone Youth's cover of "Addicted to Love" (recorded in a karaoke booth).