A population disappeared
The distinctive Neanderthal archaeological and skeletal record fades around 40,000 years ago. That is a real extinction of a long-lived population lineage in Europe and western Asia.
But the groups met and had children
Ancient and modern genomes show repeated gene flow between Neanderthals and Homo sapiens. A major shared admixture pulse in the ancestors of most non-African people occurred roughly 45,000–49,000 years ago.
The legacy is patchy
Present-day non-African genomes typically contain about two to three percent Neanderthal ancestry, but different people carry different fragments. Across a whole population, much more of the Neanderthal genome survives in pieces.
Assimilation was only part of the ending
Interbreeding cannot by itself explain every regional disappearance. Demography, climate, competition, disease, population size and absorption probably interacted differently across time and place.
Research record
Nature (2024), timing and impact of Neanderthal gene flowPeer-reviewed · open sourceDid Neanderthals Vanish — or Become Part of Us?
Direct answer: Neanderthals vanished as a distinct population, but they were not erased without descendants: part of their genetic ancestry survives in modern humans.
Extinction and assimilation both happened. The evidence rejects the simple choice between them.