I recognize that most bands can swap out their drummer or bass player without changing the essence of the group, but once you start replacing significant chunks of personnel, maybe you no longer have the original band.
Here are some rock bands that kept using their original name after some crucial pieces left:
Yes without Anderson, Wakeman, or Bruford
Last great album: Close to the Edge
Continuation story: Chris Squire owns the name, so they have kept on putting out albums through many, many different configurations.
Led Zeppelin without John Bonham (whose death precludes much touring) or Robert Plant (who is still alive)
Last album: In Through the Out Door
Continuation story: After Bonham died, Led Zep went a long time without playing together, but they have recently announced that they plan to tour without Robert Plant.
Pink Floyd without Roger Waters.
Last great album: The Wall
Continuation story: David Gilmour and Nick Mason put out a couple more albums without Waters.
The remaining people in these bands don't (or didn't) seem to understand the nature of their bands' success.
What other bands have I left off this list?