's excellent write-up Homo neanderthalensis
for a fuller discussion of the creature. See my Shanidar
for more on the flower burial that seems to indicate culture. Here I just want to address the name.
Scientists have debated whether it should be a species Homo neanderthalensis or a subspecies H. sapiens neanderthalensis. Presumably this, like the question of whether modern humans have any Neanderthal ancestry, is still open.
The name is from the Neander Valley (German Thal = valley) where they were first discovered, in 1856. The taxonomic name Homo neanderthalensis was given by William King in 1864. But in a spelling reform of 1904 the extraneous H in the German was done away with and the place became the Neandertal. This does not affect the spelling of the species or subspecies names already proposed, because original taxonomic names are immutable (though still subject to reclassification, of course). But in recent years there has been a certain trend to write 'Neandertal Man' in English. This rationalization in line with modern German spelling was first proposed by Henri Vallois in 1952.
The Neandertal, the valley itself, was named after a seventeenth-century divine, Joseph Neumann, who lived in the area. As was the common custom of the time, he published (he wrote hymns) under his name translated into a classical language. Both the Greek Neander and the German Neumann mean, of course, 'new man'.