Neanderthals are a type of ancient Humans.
Species or Race
Those who study Anthropology seem to tend to classify Neanderthals as a separate Species from us. (Believing that all modern day humans are Homo Sapiens.) However, I believe that if Neanderthals existed today (as a separate group from us), they would considered another race -- another Racial Group.
That doesn't make the label of Species an invalid one. However, one should keep in mind that if Anthropologist applied this concept of Species to Humans today, then they'd probably declare that today we have more than one Species.
Prejudice Against Neanderthals
There is a degree of Prejudice against Neanderthals. They are usually portrayed as being primitive, unintelligent, and were more like Chimpanzees or Gorillas that us. However, based on the evidence we have, this couldn't be further from the truth.
Archaeological evidence shows that the technology and culture of Neanderthals was much more advanced than that of Homo Sapiens at the time. (And thus it was Homo Sapiens who were more primitive than Neanderthals.)
Fossil evidence also suggests that, on average, Neanderthals were much more intelligent and much stronger than Homo Sapiens.
Where Did They Go
The popular theories today of where Neanderthals went, is that they simply all died out. However, the evidence we see suggests that Neanderthals and Homo Sapiens lived in the exact same region for thousands of years.
I've observed that there are people who will have sex with anything. (Animate or inanimate.) So, if Neanderthals and Homo Sapiens were living side by side, then they were probably having sex together. And thus, if they could produce offspring together, then they probably did.
Today, there seems to be quite a bit of evidence that suggests that modern day Humans are a mix of (at least) Homo Sapiens and Neanderthals. (There have been human fossils found that seem to be hybrids of Neanderthals and Homo Sapiens. Also, modern day Mitochondrial DNA studies point to Hybridization as well.)
Also, some evidence suggests that some of the characteristics found in modern day European and Middle Eastern humans may be from the Neanderthals. Such as pale skin color, light colored hair, as well as others.
Unknown Neanderthals in History
When Archaeologists find a site of ancient human activity, they seem to automatically attribute it to Homo Sapiens. And don't even consider that it could have been Neanderthals.
When looking at history, it's important to remember we are Neanderthals as much as we are Homo Sapiens. And that some of the ancient stories and history we have could have been from the people we today call the Neanderthals.
Given the maps I've seen of where the Neanderthals lived -- parts of Europe and the Middle East -- and that it is believed that the Garden of Eden is believed to be in what is today modern day Iran, I've wondered if we would have considered Adam and Eve and other people in the religious text to be Neanderthals.