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 the North Rhine-Westphalia.
So, why the spelling change? It does change the pronunciation of the word.