I like Dinosaur Jr a lot but I find all their stuff sounds better when played acoustically by J Mascis as evidenced on his album Martin + Me.
Years ago (when they still played a lot of music), MTV had an acoustic show called, IIRC, “Unplugged”. Rock and pop artists would show up and play some of their works acoustically or at least with minimal electronic assistance.
I was watching it one night and saw Aerosmith perform “Dream On”, and for the first time it hit me that these guys were really musicians, not just marketing-manufactured rock stars. Not only was the song acoustic, it was very restrained and sensual. I thought it was brilliantly better than the recorded version. Once you take away the screaming and the fireworks and the prancing, it’s easy to see who knows how to write a song.
I went from sneering at Aerosmith to liking them a lot after that.
I didn’t care abuot Aerosmith until I saw their music video with Run DMC for “Walk This Way”, and now I love them just for that video. 
One acoustic version I like is Days Go By by Dirty Vegas. I can’t say that I like it better than the techno version, but I like it a lot in a different way.
The acoustic version of Hey Ya!! 
S^G
Great, no doubt. And, then, there’s Johnny Cash’s cover (sorry, no video). FWIW, this is the version that hits me more deeply.
Oh, God, no. Just…no. I can’t stand that man’s drunken wino singing. Give me Cave any day. Same goes for the NIN “Hurt” cover.
Just realised I should have replaced “Who” not “Knows” when I edited the original post - it’s The Road To God Knows Where.
A lot of turn-of-21st-century stuff was poorly or over-produced and so despite the fact the musicans were competent the actual stuff on the radio sounded bad. But I was surprised when I heard acoustic stuff from Incubus, Matchbox 20, and (IIRC) Seven Mary Three that were actually good: they were good musicans they just apparently weren’t in the right genre.
Sounds interesting…what album’s that on?
Speaking of which…the acoustic version of Come and Get My Love from The Postman
Along these lines, my favorite cover ever is Jonathan Coulton’s acoustic Baby Got Back (starts around :30)
IIRC, it was Bon Jovi’s acoustic performance of Wanted Dead or Alive at one of the early VMAs, I believe, that started the whole ‘Unplugged’ movement (and the show).
George Harrison’s While My Guitar Gently Weeps. The original is great. The acoustic version (from the Beatles Anthology) is better.
I like Stipe’s acoustic “Losing my Religion”.
The acoustic version of “Times like these” by the Foo Fighters is also generally considered to be better than the original. I like them both, though.
If we can count those, though, Ray Lamontagne does a stellar version of “Crazy” by Gnarls Barkley.
A local radio station had Nonpoint in the studio, they did an acoustic version of Bullet With A Name, that is really great.
Another vote for Colin May. I loved it.
Some acoustic punk songs that turned out well (sorry, no links as I’m a work. You can find them on youtube though).
Tony Sly of No Use for a Name - Justified Black Eye
Bad Religion - Infected and some other song, the name of which I can’t remember.
There’s a few Foo Fighter songs that are great acoustically.
10000 Maniacs MTV Unplugged
I’ve still never heard the original, but since I can’t imagine a better version of that song, I concur. 
My vote is something close - his acoustic version of Down Under
I gotta disagree on Such Great Heights, although I heard the I+W version only once. TPS’s singing is great, and the hook is one of the best in the past 10 years, which you can’t replicate acoustically.
The acoustic version of ‘Heart of the Matter’ that Don Henley did on ‘Unplugged’ completely blew me away. Slowed down, more thoughtful, it was a much better take on the song than the original release.
The version on the “Hell Freezes Over” tour, while not completely acoustic, is based off the ‘Unplugged’ version.
Bruce Springsteen’s “Thunder Road.” One of my favorite songs ever.
Dunno if this is quite what your looking for, but Tori Amos does an absolute amazing version of “Professional Widow” (Merry Widow version) on the Hey Jupiter EP. It’s much simpler and more stripped down than the version on Boys For Pele and it just punches me in the gut every time I hear it.