I have no idea about the religions or politics or other activities of Neanderthals. But recent discoveries of cave art by Stone Age people, Neanderthals over 40,000 years ago, tell us that of all human activity the creation of art is the most fundamental and persevering. Whether in the caves of Grotte Chauvet in France, or in the cave, El Castillo, in Spain, scientists using modern methods of time-testing of antiquities, especially dating cave surfaces, have concluded that these expressions of art can be demonstrated as symbolic evidence of the societies of those prehistoric humans.
I'm not an anthropologist, but these findings tell me that of all the human endeavors, the most atavistic and fundamental, and the most revealing of all endeavors in all times, is art. Art endures, and we are fortunate it does.