There have been various theories as to why Neanderthals went extinct. A new theory injects disease as a downer.
The lineages of Neanderthals and nascent humans diverged 800,000–500,000 years ago, with Neanderthals in Eurasia and early humans in Africa.
Migrating out of Africa, humans reached the Levant tens of thousand of years before their diaspora throughout Eurasia. Meanwhile, Neanderthals never went south of the Levant.
Though humans and Neanderthal both occupied the Levant for many millennia, there was little interbreeding. Something kept Neanderthals and humans in the Levant apart. Lethal diseases for one species or the other might explain the abiding separation, as perhaps would distinct cultures, and that the 2 species were at the edge of breeding capability. What is known is that relatively few hybrids that survived had immune defenses which were carried forward genetically.
Disease probably does not fully explain how Neanderthals were wiped out. Not all Neanderthals in Europe were likely to have been put down by human diseases.
~55 thousand years ago, Eurasia weather began to swing wildly between frigid and temperate in short cycles lasting mere decades. The cold snaps brought ice sheets to what had been forests.
Animal and floral populations upon which Neanderthal fed were devastated. Nonetheless, Neanderthals, though diminished in numbers, adapted to the early rounds of climatic calamity.
The weather got worse. Oscillations between warm and frigid became increasingly wrenching. The fluxes brought profound ecological shifts in both flora and fauna.
The climatic see-saw became too much of a challenge for Neanderthals. Already splintered into small groups, they dwindled and died out ~40 thousand years ago. The timing coincides with a cataclysmic volcanic eruption in Italy, which provoked an abrupt cold spell throughout Europe.
References:
Ishi Nobu, Spokes 3: The Elements of Evolution, BookBaby (2019).
Gili Greenbaum et al, “Disease transmission and introgression can explain the long-lasting contact zone of modern humans and Neanderthals,” Nature Communications (1 November 2019).
“Neanderthals may have gone extinct because of human tropical infections,” Sci News (8 November 2019).