I recently learned that the Aerosmith song "Let the Music Do the Talking" was previously recorded by The Joe Perry Project, but since Joe Perry was in the band when Aerosmith rerecorded the song, I'm reluctant to call it a cover. Other such examples come to mind where a certain artist or act rerecorded one of their own songs, whether that was with the same band, as a solo act, part of a different band; with a new title and lyrics or just a radically different version — or any combination thereof:

• Sting recorded a dramatically different version of The Police song "Shadows in the Rain" from Zenyatta Mondatta for his debut album The Dream of the Blue Turtles
• Metallica recorded the Dave Mustaine-penned "The Mechanix" for their No Life 'til Leather demo (thanks, ted262), then as "The Four Horsemen" after they kicked Mustaine out of the band. Mustaine later recorded "The Mechanix" with Megadeth for Killing Is My Business...And Business Is Good!
• The music to Pearl Jam's "Footsteps" is the same as that of "Times of Trouble" by Temple of the Dog
• Guns N' Roses recorded an acoustic, slowed-down version of Appetite for Destruction's "You're Crazy" for their Lies EP
• Midnight Oil have a slower version of "Burnie" from Place Without a Postcard as a hidden track on their live album Scream in Blue
• Eric Clapton recorded a fast and a slow version of J.J. Cale's "After Midnight"
• Natalie Merchant recorded the folk song "Sally Ann" both as a member of 10,000 Maniacs and as a solo act

Any other examples?

"Do Ya" - by The Move & ELO

Huh. I know that song. Not sure which version I've heard, though.

Would you count the songs that The Beatles rehearsed during the Get Back sessions but ended up releasing on solo albums?

Yes, those definitely count.

Most of Bruce Springsteen's catalog.

Eddie Vedder did a version of Pearl Jam's "Can't Keep" from Riot Act for his solo album Ukelele Songs.

Eric Clapton's radically different "Layla" for Unplugged compared to the original. Does that count?

Can't think of any of the top of my head, but there are a lot of bands that have had new recordings on soundtracks, but by the time the song appears on an original album, it's been reworked. Now every time you see the movie, you hear the weird version!

Oh, here's a good one. Whitesnake's earlier, bluesy "Here I Go Again" didn't really hit until they re-recorded it as a power ballad (and changed "Like a Hobo" to "Like a Drifter" because the former sounded too much like "Homo")

But this actually is a cover song.

Yeah, I would consider Unplugged versions to count, especially if they were significant hits.

Never knew Whitesnake redid that song. Interesting!

I know (I credited the original artist). But Clapton did two versions of it. It's not a requirement for the song in question to be an original.

If I'm not mistaken, Coverdale was the only member in both versions of the band.

Only the ones that have been officially released. So... "Teddy Boy".

I'd also count the group demo of "Junk" that they recorded in 1968, and "Real Love", a demo of which was first released on the soundtrack to Imagine... John Lennon before Paul, George & Ringo used a different demo to record the Beatles' version.

I wouldn't count the Anthology version of "All Things Must Pass" simply because it's only George on the recording.

The Grateful Dead had quite a few songs that started as songs on Jerry Garcia or Bob Weir solo albums.....(though the Bob Weir solo album, Ace, was really a GD album)

Including......
Deal, Birdsong, Sugaree, Loser, To Lay Me Down, The Wheel, They Love Each Other, It Must've Been The Roses, Comes A Time, Greatest Story Ever Told, Black-Throated Wind, Looks Like Rain, Mexicali Blues, One More Saturday Night, Cassidy

Is that the only song from the Get Back sessions to have made it to Anthology?

There's also the Beatles version of "Not Guilty" which was later rerecorded by George.

Oh wait, now I get it. Duh, I'm stupid.

Anyway, yeah, it counts if an artist re-covers a song.

"White Rabbit" and "Somebody To Love" - Great Society.....then Jefferson Airplane

No, but it's the only one (on Anthology) that was later released by one of the group as a solo recording.

Ah, I see.

10,000 Maniacs did an acoustic version of "Gun Shy" from In My Tribe and released it as a B-Side to "You Happy Puppet" (from Blind Man's Zoo).