What if Neanderthals didn't simply disappear (euphemism for extinction) but were rescued by aliens when Earth's conditions began to change?
The aliens were looking at Earth because it very nearly met their conditions for a liveable planet. They encountered the Neanderthals and found them and their culture far more highly developed than that of humans. But Earth was changing. The aliens could stop the changes, but there was an ethical dilemma. Stopping the changes would save the Neanderthals, but endanger human development.
The solution was to explain the dilemma to the Neanderthals and offer them transport to another planet, one that had been lifeless but which the aliens had terraformed and seeded with life to suit themselves, and which would equally suit the Neanderthals.
Consequently, one day the aliens who return to us might be long-distant Earthlings -- the Neanderthals.