Neandertal and Neanderthal are both technically correct spellings. The Neander is a river and “Tal” is the German word for valley. But when the first fossil was discovered in 1856, the word for valley was spelled “Thal”. Neanderthal man was named for where the fossil was found – in the Neander valley. (Just like Clovis man was discovered near Clovis, New Mexico.) In any case, there were numerous spelling changes in German in 1904 to make spelling and pronunciation more consistent. At that time, the spelling of valley was changed from “Thal” to “Tal”.
So you have your choice. At the time the name was coined, it was spelled “Neanderthal”, so it would be quite correct to use that spelling. But since it’s a German word, it would also be correct to use the current German spelling: “Neandertal”
That's interesting Z-Bube. I knew people pronounced it both ways but in undergrad, the professors always pronounces in TALL. I assummed they knew their stuff. Seems you do too.