I suspect that some Neanderthal groups were more advanced than the modern public believes. They surely had group hunts for large, dangerous animals and tackled some separately, perhaps in tests of courage or for religious reasons. Like fighting a cave bear might be a ceremony... Jean Auel may not have been too far off in her books about Ayla, like, "Clan of the Cave Bear." Pretty good movie, too. I liked the haunting music.
Auel consulted with a lot of paleoanthropologists as she wrote. This does not endorse some of her later books, where I feel that the descriptive detail of such things as mammoths breeding is in poor taste. Just excessive and "over the top." I quit reading her books after I had a fill of that stuff.
BTW,I've seen actual artifacts from Lascaux and other Euro caves, at a display in Texas years ago. I was fascinated to personally behold items that I'd seen in books. Of course, that art and the other items were of Cro-Magnon origin.
I've also met Dr. Donald Johanson, whose expeditions discovered the Australopithecine skeleton they dubbed, "Lucy." He very graciously autographed my copies of his books, although he was nervous about a death threat phoned in by some nut who disagreed with his stand on evolution.
Campus cops were everywhere and I asked one sergeant why. She told me about the death threat. (SMU police are sworn officers; not guards.)
My teen daughter found Dr. Johanson to be fascinating, too, and was far more interested in his presentation than she'd expected to be. He did a good narrative, with an outstanding slide show.
Of course, he usually writes about times long before even Neanderthals, but if you study earlier human-like forms, you can see the progress.
Some feel that the palate in Neanderthals wasn't shaped right for more complex speech and they may not have had the creative vision of newly arrived Cro-Magnons. They went for thousands of years relying on clubs and spears. I don't think they ever evolved the atlatl or the bow. Or the wheel.
The prevailing theory is that Neanderthal features were an adaptation to living in extreme cold. As the Ice Ages faded, they may have to some degree just evolved out of that mode. But I think their genes were distinct; they could breed with modern men but were not the same species, really.
Interbreeding may have weakened their gene pool and produced mixed humans. But I suspect that Cro-Magnons just wiped out those less advanced people, for the most part.