what’s a song cover that you think is better than the original?

I think Dionne and Burt are a wonderful combo, but I don't have any records by them, whereas I do have every Stranglers album featuring Hugh Cornwell. Very different indeed, but personally I wouldn't say either is 'better' - vive la differance! 

Nirvana, Where did you sleep last night, is a favorite cover of mine. Counting Crows Like Teenage Gravity is a great cover as well. Easily the best sounding cover I own is a pressing of Kat Edmonson's version of Summertime from her Take to the sky album. 

Cream's version of Robert Johnson's 'Crossroads'

and Willie Dixon's 'Spoonful'

Santana's version of Fleetwood Mac's 'Black Magic Woman'

Pete Seeger and other's versions of Woodie 

Guthrie's 'This Land is Your Land'' as long as ALL the lyrics are sung

And one of the best covers ever is ... Jimi Hendrix's of Dylan's 'All Along the Watchtower'

IMHO it's uncommon for a cover to overtake an original, 

but they do expose new audiences to not just unfamiliar songwriters, but for the curious often different genres of music also.  

Grace Jones - In Crowd

Randy Crawford - Imagine

UB40 - Ain't no sunshine

Joe Cocker - A little Help from my Friends

Hendrix  All Along the Watch Tower