Ever since I first seriously studied Neandertals (yes, that's the correct spelling) as an undergrad, I have had this rather sad, poignant image in my head:
It's somewhere between 24,000 and 30,000 years ago. It's night time. There's a young man sitting alone by a small camp fire fire. I always imagine him as young but an adult because someone who is old enough to have learned survival skills and yet young enough to be able to fight off disease and starvation is the one who is likely to have survived. I imagine him as male because I am male, and therefore have an easier time putting myself in his position if he is as well. He is the last Neandertal on Earth.
His band is gone, except for him. This band may have included his parents and siblings, and likely included his mate (assuming, and it is an assumption, that they were given to monogamy) and children. He's the last one, though. The others may have died in a violent conflict with the other creatures, the tall and thin ones with the high foreheads who are the ancestors of you and me, or they may have died of starvation due to a changing climate, or they may have simply been pushed away from their normal food sources by the other creatures. It doesn't matter now, because they are gone and only our young man remains, and he is starving to death.
He sits by his fire not knowing that he is the last of his species. Bands as small as his was are fluid, members come and go, and maybe he expects that he will find more of his kind and life will go on, should he survive.
Is this pondering likely to have been true? Of course not. The last Neandertal is just as likely to have been female as male, may have been a child or may have been elderly. They may have met a violent fate at the hands of an anatomically modern human (our species) or a wild animal. There is some evidence to suggest that Neandertals could have interbred with anatomically modern humans and therefore there may not have been a last Neandertal, but rather an elimination of the species as their genes were spread within (or phased out of) the Homo Sapien sapien gene pool.
Stories like this one about the last Neandertal occupy a weird place in archaeology. On the one hand, there is rarely data to support them, and therefore they are on shaky ground from the start. They often create images in our minds that can interfere with our ability to judge data more clearly. At the same time, these sorts of stories are often what keep us interested in our work and prevent us from making the mistake of thinking of past people as little more than statistics.
The Neandertal vanished somewhere between 20,000 and 30,000 years ago. We don't know what their last days were actually like, whether they vanished slowly, were victims of violence, or simply integrated into the anatomically modern human population. The last unequivocal evidence we have for them is from southern Spain, and it dates to approximately 25,000 years ago. In the same cave that holds this evidence, there is cave painting associated with anatomically modern humans that dates to approximately 20,000 years ago. Whether we pushed them out or they simply faded away, we ultimately took their niche.