Neanderthal or Neandertal?

Both of these spellings (and pronunciations) are commonly found - is either more correct to use?

In short, nope!

Neanderthals get their name from the Neander Valley (or Neandertal in German), near Dusseldorf in Germany.  This valley is where the first Neanderthal fossils were found, around 150 years ago.

OMG, it's a Neandertal baby!!
Or is it a Neanderthal baby...?

'Thal' is an old form of the word 'valley', so Neandertal (the place) used to be called Neanderthal.  Nowadays 'thal' is spelled as 'tal' after a German spelling reform at the beginning of the 20th century (people loved those back then, didn't they?), which is probably why we see Neandertal in spelling now and again.  It tends to be less
common I get the impression...

Neanderthals' taxon is Homo neanderthaensis, with the 'th', so I guess purists could argue that it's better to spell it with the 'th'.  But what is appropriate in language is defined by what people use (generally), so whichever!

'Thal' and 'tal' in the German are both pronounced the same, which reflects how some people (like me!) spell Neanderthal but say 'Neander-tal'.  However, pronouncing the 'th' is totally common and often used in the English pronunciation of the species.

In short, say whatever, spell whatever... it's language!  As I said, what is 'appropriate' is all about what people use, and in what situation.  It's all about conventions, and there are conventions of using both :)