I have heard both all my life and it drives me buggy. I have a sneaking suspicion that it’s both but I don’t have a link to an online dictionary nor do I have a hard copy here at home so I am asking the TM’s.
If you want to show you’re “educated,” “erudite,” and possibly some other “e-” words, it’s “-tal,” which preserves the German pronunciation.
If you’re talking to Joe Blow, pronounce it as the familiar “-thal.” That way they won’t think you’re putting on airs.
aha, you may want to bookmark this url as it is rather obscure: http://www.dictionary.com 
Seriously, though, it is either. An anthropologist will scoff at Neander “thal” but your average guy on the street still pronounces it this way.
If I’m not mistaken, Neander “tal” is relatively new, even in the world of Academia. There is, however, a good chance that I am mistaken.
New York Times decided to go with Neanderthal. C. Loring Brace decided to follow suit and so what the heck - Neanderthal. But I bet Brace says “…tal”
There you go, aha: From Merriam-Webster
It’s the same way with “Celtic”. If you pronounce it “Keltic” like the academics and people on PBS do, Joe Blow doesn’t know WTF you’re talking about. But if you pronounce it “Seltic”, Joe Blow will punch your lights out, 'cause he thinks the Celtics suck.
The word “thal” is German for valley; akin to the Swedish “dahl” and the English “dale or dell”. Since the first bones were found in the valley of the Neander River in Germany, it’s “Neanderthal”. The problem is, the Germans pronounce the word “tall”. So “Neandertal” is an attempt to keep the proper pronunciation by changing the spelling, like “Keltic”.
Just to expand on what other’s have already said:
The first remains of homo sapiens neanderthalensis were discovered in 1856, in the Neanderthal valley of Germany. Germans don’t have a theta (th) sound in their language. If it’s written th, they pronounce only the t.
Most European academics always referred to it as “tal”, but American academia somehow missed out on that and naively pronounced it as “thal”. (We got Beethoven right, at least.) By the time the American intelligensia got it straight, it had trickled down to hoi polloi with the incorrect pronunciation. So what Mjollnir and TNTtruth said goes.
I’m all fizzy because you didn’t plunk an English article in front of the Greek article “hoi.” I am SUCH a geek.
Anthros
All information I could have found on the internet if I weren’t so lazy…
The valley was spelled Neanderthal at the time of the skull discovery and made it into the literature using that format. (Note that the Latin version still has the “h”.)
The Germans normalized their spelling to eliminate the “h” in the valley’s name earlier in this century, and the scientific community has slowly been converting to that spelling ever since.
A ha! I knew I could find this somewhere!
In talkorigins I found:
Neanderthal or Neandertal?