Tom Tom Club was basically Talking Heads without David Byrne.
Hindu Love Gods was R.E.M. with Warren Zevon instead of Michael Stipe.
Any other bands that are essentially the same as another band except for adding/removing/replacing one person?
Any of the variations on Crosby Stills Nash (and Young).
Audioslave was Rage with a vocalist swap
New Order was Joy Division less Ian Curtis
Van Hagar was Van Halen with both Van Halens, but less one Roth…
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Badfinger was the Beatles minus John Lennon.
When Robert Gotobed left the band Wire, the three remaining members of the band dropped a letter from the band’s name and called themselves Wir.
Bow Wow Wow was Adam and the Ants with a 14-year old girl lead singer instead of Adam Ant. (Although Adam quickly assembled new Ants).
I don’t know if this quite counts, but Wild Flag is 2/3 of Sleater-Kinney, but other than swapping Corin Tucker for Mary Timony, they also added Rebecca Cole on keyboards. So, almost a swap, but also an addition.
Not quite, but close:
Liquid Tension Experiment was an instrumental band consisting of John Petrucci and Mike Portnoy of Dream Theater, Jordan Rudess, and Tony Levin. After two albums, Rudess joined DT, and the decision was made to end LTE. If LTE had gone on, it would have been DT with a different bassist and no singer.
geto boys was nwa without eazy e or dr dre.
Love and Rockets were Bauhaus without Peter Murphy.
Gillian Gilbert wasn’t in Joy Division. Still counts for this thread, though, I think.
The Mission was initially The Sisters Of Mercy with a drummer instead of Andrew Eldritch.
Led Zeppelin was basically Led Zeppelin without John Bonham.
The first incarnation of Ritchie Blackmore’s Rainbow was the same as Ronnie James Dio’s band Elf, with Blackmore replacing the guitarist.
Velvet Revolver was Guns n Roses with Axl swapped out for two other guys whose identities I’m blanking on.
Technically these fit the OP, I guess:
The Band was Bob Dylan’s band (on tour), without Bob Dylan
The Eagles were Linda Ronstadt’s band (on tour), without Linda Ronstadt
Well, legally/technically, Led Zeppelin were The Yardbirds - Jimmy Page retained legal rights to the name when the band disbanded. When he replaced members and got JPJones, Plant and Bonzo, they first toured as The New Yardbirds before taking the name that Keith Moon mentioned when he stated how he thought Page’s music ideas would go over…