Neanderthal is the more common spelling of the noun denoting the species of robust humanlike creatures that went extinct around 30,000 years ago. Neandertal is preferred by a few scientific publications.
Neanderthal, the original spelling, was derived from the German valley where Neanderthal fossils were first discovered in the 19th century. In 1901, however, the German name of the valley was officially changed to Neandertal. Some scientists and scientific publications have extended the change to the name of the species. Most have not.
Nonscientific publications show a similar trend. Neanderthal is preferred, but Neandertal appears about a quarter of the time. Neither spelling is inherently correct or incorrect, but some scientists do have strong feelings on the matter. If you’re writing for a professor, find out what he or she prefers.
Scientific American is one scientific publication that favors Neandertal (although they do use Neanderthal occasionally):
Paleoanthropologists know more about Neandertals than any other extinct human. [Scientific American]
But Scientific American is in the minority. Most popular scientific publications, such as the ones below, use Neanderthal:
Carbon-dated Neanderthal remains from the foothills of the Caucasus Mountains suggest that the archaic species had died out before modern humans arrived. [Nature]
Speaking of ancient makeup, a new find suggests that Neanderthals wore “body paint” 50,000 years ago. [American Science]
Neanderthals dominated Europe for some 200,000 years until modern humans began moving into the region about 45,000 years ago. [National Geographic]
In the cooler regions occupied by Neanderthals, heat exhaustion would not be a problem … [New Scientist]