Late to topic again!! Songs are usually written for a voice, so anyone else singing it will sound odd which is why covers often are re-arrangements of the original etc (and not just a carbon copy with different musicians), sometimes it works sometimes it doesn't.
As already stated by others:
Johnny Cash - Hurt, I just think it flows far better than the original and the video for it just adds extra weight to the track, it's haunting.
Turisas - Rasputin, much better than the original. Seen it live twice and it's fun as hell.
Judas Priest - Green Manalishi.
Some songs have been covered sooo much though...but, the title of the thread is covers better than original not "best cover versions"
Joan Jett & The Blackhearts - I Love Rock'n'Roll,
Original .. which actually is decent enough on its own, even though most people have never heard it.
Kiss - God Gave Rock and Roll to you II,
Original. As much as Kiss have several 'problems', I do prefer their version over the original, different lyrics too. However the best version is the Bill & Ted one with Steve Vai's intro solo included... not sure if it's even been commercially released though :/ (
http://www.youtube.com/watch?v=kfcS9yRzYwg)
Rainbow - Since you been gone,
Original - Maybe it's just nostalgia, but the Rainbow one just sounds so much better to me.
Judas Priest - Diamonds and Rust (acoustic), I know it's been mentioned already but they did two versions, and although it's a close call between which one I prefer - their electric cover or their acoustic one, I think this one takes it as its a bit more emotional.
Guns N Roses - Live and Let Die,
Original, just better in my opinion.. plain better. Although I am biased as GnR are one of my favourite bands (even Chinese Democracy, yes).
Guns N Roses - Since I don't have you,
Original, Although I (shockingly) actually quite like the original.. it's not something I'd ever put in a playlist, but am happy to listen to if it comes on the radio. I won't go any further into the rest of the Spaghetti Incident as a lot of them are pretty good.
Glenn Tipton - Paint it Black,
Original, I love the original for this but I love the Tipton version too.. and it makes it into more playlists than any of the other versions I have (All of which I love - Hayseed Dixie and W.A.S.P being the other 2, all are different styles).
W.A.S.P - Somebody to Love,
Original, Okay this is more just a matter of me liking metal I guess.. but I do think its a good cover. I won't list any more WASP covers as they've done quite a few... The Real Me, Paint it Black, Burn, Mississippi Queen...
There are a lot of songs I think are equally as good as the original for various reasons and would really struggle to call either one "better"
Northern Kings - Don't Stop Believing,
Original, I love the original (although rarely listen to it anymore due to events) and I wouldn't nessecarily say the Northern Kings version is "Better", but it's fantastic in its own right.
Northern Kings - My Way,
Original - Love the NK version! But its more my style so thats only natural.
I don't class them as covers..but Halfords versions of Prisoner of your eyes and Heart of a Lion are superior to the Priest released recordings IMHO.
Okay, I think thats enough, I can't be bothered finding youtube links anymore
... Johnny Cash did a lot of covers and many of them are fantastic. (Ring of Fire is a cover
), His cover of U2's "One" and Neil Diamond's "Solitary Man" are both superior imho, whereas his cover of Will Oldman's "I see a darkness" is fantastic, but I would struggle to call it better than the original I find them both good for different reasons.
Hayseed Dixie I think deserve a mention all on their own, their covers aren't nessecarily better than the original because it is a COMPLETELY different style, but I do enjoy a listen to it, it certainly puts you into a more upbeat mood. I saw them live a few years ago at Donnington, there was more people at the small tent watching them than were at the main stage watching MCR.. who left early due to being bottled.
I need to stop writing essays.