How should we pronounce it - Neanderthal or Neandertal?

The first such fossil was discovered in 1856 in the Neander Thal, or "Neander Valley" in German, and became known as "Neanderthal Man". In 1904, German spelling was regularized to be more consistent with pronunciation, and "thal" became "tal".

In 1952 Henri Vallois proposed that it should be spelt as the Germans spell it, and the "-tal" spelling has become widely used since then. The "-thal" spelling persists most strongly in England. However you spell it, it should always be pronounced with a 't' and not a 'th' (German has no 'th' sound).

None of this affects the taxonomic name of the Neandertals. William King proposed the name neanderthalensis in 1864. Since then, opinion has fluctuated as to whether they should be considered sapiens neanderthalensis (a subspecies of Homo sapiens) or a separate species, neanderthalensis. For the first half of the 20th century, they were usually considered a separate species. For the last few decades they have usually been considered a subspecies, but recently neanderthalensis has been gaining in popularity again.

In either case, the 'h' must remain in the name, because the laws governing biological nomenclature forbid changing the spelling.

The Neander Tal was named after a minister, Joachim Neumann, who used to take walks there in the late 17th century. Neumann composed many hymns, some of which are still sung today. Wanting to use a Greek pseudonym, Neumann, whose name means "new man", chose "Neander", a translation of his name into Greek. By a strange coincidence, the "New Man Valley" named for him after his death gave its name to a new type of human that was discovered there.