Will Neandertals be able to advance beyond Mousterian? I think it'd be an interesting what-if situation if they managed to stay around past their historical extinction and develop more.
There are two problems with this.
First, Neanderthals really did not progress beyond the Chatelperronian so even if I wanted to make up new cultures I would have no idea what to call them or what they would look like. Second, all research indicates the Neanderthals really were lacking something that we didn't - they seemed to have no ability to make representational art or think in the abstract. They could make great tools all right and even make some decorative art but by all accounts they had no ability to form religion or think beyond the here and now.
In game the Neanderthals may appear to be doomed but that's not necessarily true. Humans start off hating them like crazy but as time passes this fades to the same sort of malus that all cultures have for foreigners. At the same time it is only Paleolithics that want to drive them to extinction - if you can survive to the Mesolithic then you will be just another culture on the map. Finally, once you get to the Neolithic and become Tribal all of the old restrictions on religion, tech and so on cease to apply, so your Neanderthal/Mousterian/Chatelperronian realm can actually become Tribal, then Nomadic, Feudal or whatever and conquer the map.
One of the main reasons I started this project was to see if I can take my Neanderthal band into the wilderness until the human holocaust had passed and then see if I could make a viable realm in the Bronze Age and beyond. I am looking forward to finishing the late game mechanics and reading about players who managed to do just that !