Are you a lumper or a splitter?

[Edwardo Piet]

Neanderthals were a separate species from modern humans, and became extinct (because of climate change or interaction with modern humans) and were replaced by modern humans moving into their habitat between 45,000 and 40,000 years ago. Jared Diamond has suggested a scenario of violent conflict and displacement.
Neanderthals were a contemporary subspecies that bred with modern humans and disappeared through absorption (interbreeding theory).
Volcanic catastrophe: see Campanian Ignimbrite Eruption

We aren't special... we're just violent and horny.

And, neanderthals were closely related enough to us to breed with us. They also made tools and even did cave art.

Furthermore, some of us still have substantial amounts of neanderthal DNA:

Since 2010, evidence for substantial admixture of Neanderthals DNA in modern populations has accumulated. Evidence of admixture was found in both European and Asian populations, but not in Africans, suggesting that interbreeding between Neanderthals and anatomically modern humans took place after the recent "out of Africa" migration, likely between 60,000 and 40,000 years ago.