Covers that are arguably better than the originals:

When you're a absolute toplevel songwriter but a very limited singer, Better singers tend to lend your songs and do it better ...

Nick Cave singing a Leonard Cohen song:
Madeleine Peyroux doing the same with an other song from Cohen
 
And in the category "you probally did not know it was a cover" i'll put this original 1962 recording by Prince Buster & His Allstars (record in video is the 1964 repress) that became word famouse 15 years later or so... But this is the original jamaican tune.