Personally, I love covers that completely change the song. I don't really get the point of listening to the same song but with different vocals, but... whatever floats your boat.
I love They Might be Giants' cover of Whirlpool by Meat Puppets and Mindless Self Indulgence's cover of Big Poppa by The Notorious B.I.G. (And yes, I know MSI sucks, I just listen to reuploads of their songs, because god dammit they have some bangers).
Love how they both basically do the opposites of each other, haha. TMBG makes it slower, and MSI makes it faster.
I'd also put Overcome by Tricky, but it's not really a cover, it's just got the same lyrics, plus Tricky was in Massive Attack, so that doesn't count. I still love it, though. Didn't even realise the lyrics were the same as Karmacoma the first time I listened to it!