But some people are more familiar with the cover by Sixpence None the Richer.
Soul Asylum’s “Runaway Train” sounds like a Tom Petty & The Heartbreakers song. While I’m not intimately familiar with Soul Asylum’s discography, nothing else I’ve heard from them reminds me of Petty, just that song.
You mean it’s not??? What the hell!
I’ve never thought about it, but now that you mention it, it’s true! But I’ve always thought that Dave Pirner is a dead ringer voice-wise for Thelonious Monster’s leader Bob Forrest. They had their heydays at the same time (my heydays too ;)), and I’ve always thought that the bands sounded like they covered each other.
For comparison:
Thelonious Monster - I Live In A Nice House
Soul Asylum - Without A Trace
“Lady Picture Show” by Stone Temple Pilots sounds like a Badfinger cover to me.
Holy cow, I never noticed that! Spot on.
Much of The Spongetones’ output sounds like covers of other groups, in this case The Hollies.
My girl Maryanne
She tries to get in as many words as she possibly can
I love Maryanne
She understands
I think that the Hollies always sounded like a cover of various other bands. I listened to a lot of music during the period that they were popular. When I, decades later, ran across their name and wasn’t sure I remembered them, I listened again to their music. Each song of theirs that I remembered I thought that some other group of the time did. I had no idea that those songs were all from the same group.
Franz Ferdinand’s “Take me Out” is a cover of Genesis’s “That’s all”
Weird Al Yankovic is the master of deliberately doing this. (He calls them “style parodies”). They aren’t parodies of any particular song, but the style of a certain artist. He usually has 2 or 3 on each album, and they are often the best songs on the album.
Examples:
Germs - Nine Inch Nails
Everything you know is wrong - They Might be Giants
Skipper Dan - Weezer
Genius in France - Frank Zappa
CNR - The White Stripes
Dare to be Stupid - Devo
Well, “Just One Look”, “The Air That I Breathe”, and “He Ain’t Heavy, He’s My Brother” are all covers. (Of these three, only “Just One Look” charted under an earlier performer.) But most of their radio singles were original compositions.
Right, they didn’t do covers in general. My problem was that when I rediscovered them in the past few years I was surprised to discover that a bunch of songs I vaguely remembered were all by the Hollies. Each of their songs sort of reminded me of the songs of other groups of the time when the Hollies were most popular. Somehow I missed noting that each of those songs was by the Hollies, since I didn’t listen closely to what the D.J.s would say as they introduced their songs. In my mind, up to the point that I rediscovered the Hollies, I had either assumed that each of those songs were by various one-hit wonder groups or were by various well-known groups who did songs vaguely in that style.
I think I’ve told this story before…years and years ago I was at my mom’s for Thanksgiving, and my brother had one of the football games on. I wasn’t paying a lot of attention, and then it was halftime. I was half paying attention, but I asked my brother "When did Pearl Jam get so bad?"
Turns out it was Creed.
You forgot Dog Eat Dog in the Talking Heads style, and Bob in the style of Bob Dylan.
Didn’t forget… just didn’t want to make an exhaustive list because I’d be here all day. 
(Craigslist - in the style of The Doors, Mission Statement - in the style of CSNY, etc., etc.)
Another Dire Straits error. First time I heard Sultans of Swing, I thought it was some new amazing Dylan track.
Nope.
Leonard Cohen’s Death of a Ladies Man Sounds a bit like Pink Floyd.
The Hollies “Long Cool Woman” sounds like CCR with the twangy guitar and lazily spoken lyrics.
And “La Grange” starts off like a John Lee Hooker “Boogie Chillun” cover.
When I was a kid in the late 70s, I thought “Sultans of Swing” was by Eric Clapton.