The Headstones cover of Traveling Wilbury’s “Tweeter and the Monkeyman.”
I strongly recommend you listen to it.Just Like a Woman - Joe Cocker
Elvis Presley - That’s All Right
Elvis Presley - Hound Dog
Elvis Presley - All Shook Up
Elvis Presley - Blue Suede Shoes
Elvis Presley - probably a lot more covers I can’t think of right now.
Blue Moon - The Marcels
Iron Maiden did a very very good job on I Can’t See My Feelings, which is kind of a toss-up compared to the original, but it is the only cover of a Budgie song that might arguably be “better” than the original.
Speaking of how Joe Cocker is feeling, today …
Dead apparently of lung cancer. What a shame. 
Someone in IRL told me she read that he died of undisclosed reason, so I thought I’d better post this BBC article saying he had been battling lung cancer
Written by Leon Russell
House of the Rising Sun - Frijid Pink. I like this version much better than The Animals. But then they are not the original artist. No link because I have a work filter, too. (I’m surprised they don’t filter the Dope. If they only knew…)
+++1
And, speaking of the Zombies, I prefer Jackson Browne’s version of Stay (particularly when segued from The Load Out.
And, speaking of Jackson Browne, I prefer the Eagles version of Take It Easy (because it doesn’t segue to something else and the harmonies are better).
Oh, and I really appreciate Clapton’s version of Marley’s I shot the Sheriff more than the original.
Sorry, no links; work blocks my YouTube access.
–G!
I’d say that SRV’s live cover of Jimi Hendrix Voodoo Chile is the definitive example of this.
Wilson Pickett’s cover of “Hey Jude”
Jack White brought insanity to ‘I Just Don’t Know What to do with Myself’, which it needed.
Incredible Bongo Band - Apache
Alien Ant Farms’ “Smooth Criminal” beats out Michael Jackson by a mile. Sorry MJ!
The 1992 version of “Don’t Let The Sun Go Down On Me”, live between George Michael and Elton John is better than the 1974 studio original. I know it is not 100% cover, only 50% due to George’s presence. On that note Elton seems like a feature artist, upstaged by Mr. Michael.
I agree that Santana’s version of “She’s Not There” beats the Zombies version, but only by a hair.
“Tainted Love” by Soft Cell is better than the original version, and other versions.
“Always Something There To Remind Me” by Naked Eyes is better than the forgotten 60’s version.
“If You Ask Me To”- Celine Dion does it better than Pattie Labelle.
“I’m Every Woman”-Whitney Houston does better than Chaka Khan.
While I am tired of it, Whitney does better than Dolly Parton on that song which was run to the ground. At least her voice is better.
Paul Young’s hit version of “Every Time You Go Away” is better than Hall and Oates, though that is good also.
Paul Young does a good job on “What Becomes of A Broken Hearted”, best version or maybe because it reminds of being a little kid in the early 90’s.
UB40 did good covers, “Here I Am” is better than the Al Green original. UB40 out does Elvis also, though I like that version.
Sugar Ray does “Is She Really Going Out With Him” better than Joe Jackson-NOT!
I do believe Madonna does “American Pie” better than Don.
(starts running for his life as Dopers chase after him).
Ryan Adams version of “Wonderwall” is good, still Oasis does it best. Speaking of that song, Paul Anka did a nice swing version of it, pretty good. He also sang a nice cover of “Ordinary World” by Duran Duran.
As mentioned Gary Jules does “Mad World” justice, the original by TFF sounds so rushed.
“Love Rollercoaster” by RHCP is better than the 70’s by Ohio Players- yet the latter has a better intro.
Here are some interesting Nirvana covers:
Patti Smith - Smells Like Teen Spirit
Lana Del Rey - Heart Shaped Box
Also,
Junior Parker - Tomorrow Never Knows
Dave Matthews - Take Me To Tomorrow
Placebo - Running Up That Hill
Not saying these are definitely better, but they are great covers.
Tori Amos does a nice cover of Smells Like Teen Spirit, too.
I prefer Triggerfinger’s cover of Lykke Li’s “I Follow Rivers” to the original. Even though he gets some of the lyrics wrong.
The Tea Party - Paint it Black
nm