I was going to mention RESPECT from Otis Redding, but Franklin’s version is not radically different enough I think.
That’s what I came here to mention. Although I would argue that Franklin’s version is sufficiently different from Redding’s. Anyway, since I was ninja’d, I’ll mention Breakaway. Originally an R&B song by Chuck Jackson, it was covered in the 80’s by Australian funk group Big Pig. It was a hit in Australia, but US audiences most likely know it best as the song played in the opening credits of Bill and Ted’s Excellent Adventure.
Bobby Darin’s hit version of Mack the Knife was very different from the earlier versions by Bertolt Brecht and Lotte Lenya.
The disco version of A Fifth of Beethoven was a hit. Ludwig van’s original was a hit in its day, but never made the Hot 100.
Hey Joe was first recorded by the Leaves:
A few years later, the Jimi Hendrix Experience recorded a version that is, understandably, much better known:
Nothing Compares 2 U.
Tainted Love - American Gloria Jones’ Motown-influenced 1964 original never charted, although about 10 years later it attracted club attention in the UK. Soft Cell’s 1981 new wave reworking was a huge world-wide hit, including best selling single of the year in the UK.
The original “Roses” by SAINt JHN is a slow-paced rap song that went to #1 in the U.K.:
The Imanbek Remix, which went to #4 in the U.S., sounds like The Chipmunk’s latest disco release:
Nothing compares 2U - The Family (a Prince side project) - Sinead O’Connor. (Actually, I’ve always been surprised these are so similar - they’re close enough that I marvel that nobody saw the greatness til O’Connor did it)
Tainted Love - Gloria Jones - Soft Cell
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My bold. This popped up when I was watching Lis Sorensen
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Haven’t read the whole thread yet, but the first song that came to mind was “Respect”. A great song, and a decent hit for Otis Redding, one of his best songs, but instantly blown out of the water by Aretha Franklin’s cover. Of course she rewrote the song and changed its meaning, but also the arrangement was far superior. Otis himself admitted that from then on, Aretha owned the song.
ETA: oh, links, I forgot. Do I really have to for this song?
There are a few hit songs that you wouldn’t BELIEVE are covers. Or maybe it’s more proper to call them “re-workings” since they’re SO different from the original. But they are nonetheless fundamentally based upon earlier songs.
Curtiss Maldoon (who?!) and their Sepheryn (what?!) became Madonna’s Ray Of Light.
Dazed and Confused was originally recorded not by Led Zeppelin but a man named Jake Holmes, who I’m sure would forgive you if you didn’t have a clue in hell who he was. (If you joined the Army during the 1980s, his “Be All That You Can Be” jingle may have inspired it - yes, he actually wrote that too.)
Damned, I swear that I did a thread search for “Respect” and nothing came up before posting. Shitty search function. 
Would “Misirlou” count? The melody is a traditional Greek folk song so I don’t know who would get credit for the “original”, but I’m sure there are were plenty of recordings of it being played in a more traditional style early on. According to Wikipedia there were a couple of jazz arrangements recorded in the 1940s. But it was Dick Dale’s surf guitar version that became a hit.
“The Lion Sleeps Tonight” was originally recorded by South African singer Solomon Linda in 1939 under the title “Mbube” with lyrics in Zulu. Later it got English lyrics and was covered by numerous artists, the best known one by The Tokens. The basic melody stayed pretty much the same between the original and the cover, the only radical difference was getting English lyrics.
Maybe referring to different values of “hit,” but a lot of the old songbook standards from the Tin Pan Alley era were utterly, wonderfully transformed by jazz musicians decades later.
“Manhattan”, from the 1925 Broadway revue “Garrick Gaieties” (1929 movie short shown here):
“Manhattan,” Blossom Dearie, 1959:
Sort of regionally specific case, and it took me a bit to decide that they were different enough to count.
Slade’s Cum On Feel the Noize was a #1 hit in the UK, but didn’t do anything worth noting in North America.
Quiet Riot’s cover was #1 in the US and #8 in Canada.
Slade:
Quiet Riot:
Came here to say this, glad it’s taken care of.
TRIVIA FACT: Gloria Jones was the driver when Marc Bolan was killed in a crash. She was probably drunk, having been drinking wine with Bolan at the restaurant they had just left. She was set to be tried, but skipped out to the US.
They’re not radically different, but, yeah, they come up in conversation about three or four weeks ago and people were a bit stunned when I said “you know, that’s a cover.” Kind of like “Once Bitten Twice Shy,” as well.
Kris Kristofferson explicitly calls his song “Me And Bobby McGee” a country song, in a spoken word into to his recording: “If it sounds country, man, that’s what it is; this is a country song. Okay, then, one, two, three, four…”
Janis Joplin sang it as a blues/rock song.