Turns out this is a hard question. You should know that before you click over to any of the news reports about the Russian "Neanderthal" site at Byzovaya that has been radiocarbon dated to 33,000 years ago, which these same news reports will tell you is 7,000 years after Neanderthals were supposed to have gone extinct. First, there are no bones or DNA from the Russian site. It was identified as a Neanderthal site by the stone tools, which belong to the "Mousterian" industry (above and below). Most archaeologists accept that, in general, Mousterian tools in Europe were made by Neanderthals, and that the change from Mousterian tools to new, Upper Paleolithic or "Aurignacian" industries around 40,000 years ago represents the arrival of fully modern humans in Europe. But there are complications, including sites that have produced both sorts of tools, tools that don't really fit into either tradition, and a couple of sites that have produced weird mixes of modern human bones and Mousterian tools, or Neanderthal bones and Aurignacian tools. And what about the interbreeding between the two sub-species, which recent DNA analysis suggests took place?
There are also numerous archaeological sites dating to later than 40,000 years ago that have produced Neanderthal bones or Mousterian tools (John Hawks runs through some of this evidence here.) And, to further complicate things, there is a major dispute about the validity of radiocarbon dates from this period; one archaeo-physicist recently argued that we should discard all radiocarbon dates from Europe for the 30,000 to 50,000 BC period that were run more than five years ago.
But all that aside, this is a very cool discovery. The site is in the Arctic, and it shows that people using crude stone tools -- likely Neanderthals -- could thrive in that environment. The site produced numerous butchered bones of mammoths, woolly rhinoceros, and other Ice Age beasts. It is a fascinating glimpse of human life in a vanished world. It also suggests that modern humans and Neanderthals lived side by side in Europe for 10,000 years, which has all sorts of interesting implications for the presence of ogres and goblins in our folklore.
Good article at Popular Archaeology here; thoughts from John Hawks here.