Same title, different song

“Creep” by Stone Temple Pilots
“Creep” by TLC
“Creep” by Radiohead

“Thank You” by Boyz II Men
“Thank You” by Led Zeppelin

“Goin’ Back to Cali” by LL Cool J
“Goin’ Back to Cali” by the Notorious B.I.G.

“Change of Heart”:
Eric Carmen
Tom Petty & the Heartbreakers
Cyndi Lauper

“Games People Play”:
Joe South
The Spinners (also titled: “They Just Can’t Stop It”)
Alan Parsons Project

“Dreams” - the Allman Brothers Band and Fleetwood Mac

“I Love You”:
People (60’s psychedelic band)
Climax Blues Band
…and many others…

“I Want You” - Rachael Yamagata and (sorta) The Beatles

“God” - Tori Amos and The Beatles

I’m sorry, that should actually be credited to John Lennon.

Crazy
Patsy Cline
Barenaked Ladies
Aerosmith
And I think Britney Spears

One
Metallica
U2
Harry Nilsson (and all the covers thereof)

Somebody To Love - Queen and Jefferson Airplane

You Keep On Knocking - Suzi Quattro and Laura someone - I have it on a '45 sided record with a small centre hole. It plays at '78.

Add Dido

I remember checking my iPod once and noticing that I had:

“Disappointed” by Bob Mould
“Disappointed” by The Connells
“Disappointed” by Public Image Limited

Add in “Disappointed Man” by Freedy Johnston and we’re talking about some serious disappointment going on!

“Jump”

Van Halen
The Pointer Sisters

More examples here

And Jamelia. And Christina Aguilera (I think).

What a Wonderful World

Sam Cooke
Louis Armstrong

And Puffy Ami Yumi (I can’t believe I forgot that one earlier. >_<)

Isn’t there one Power of Love for every star in the Milky Way?

The Power Of Love

Huey Lewis & the News
Frankie Goes To Hollywood
Jennifer Rush / Celine Dion

I really didn’t see you before I posted.

Leon Russell’s first album contains a song titled “Give Peace a Chance” that is not the Plastic Ono Band song, and a song titled “I Put a Spell on You” that is not the Screamin’ Jay Hawkins standard.

“Saving all my Love for You” Whitney Houston & Tom Waits

Add Dannii Minogue.