This is, of course, very subjective, but sometimes a clever cover comes along that makes you say, "That was better than the original." List some of your favorites here:
"Take on Me": Reel Big Fish > A-Ha
The staccato horn section gets me every time
(the song that got this whole train started in my head; I introduced the RBF version to Younger Offspring, and every time we hear any version of this song, we agree about this assessment).
"With a Little Help From My Friends": Joe Cocker > The Beatles
The entire change in sound, and the background singers
(honorable mention)
"Wonderwall": Ryan Adams >= Oasis
I go back and forth on this one. The RA version is totally "emo", but it gets me in "the feels" (as the kids say) sometimes.