I'm aware that it's accepted Neandertals might've been as capable in many areas of survival, however I'm not aware that it's been accepted they had more advanced brains. Got a source?
I remember hearing somewhere, one source I remember [1], that despite their intelligence, they tended to rove in smaller tribes partly perhaps due to less inter-communication skills, which may've led to a disadvantage in warfare.
Or perhaps, my own thoughts, the 'loner genes', for lack of a better term, iterated more throughout the Neaderthal ancestry which led to smaller tribes. Another guess, maybe forested Eurasia, which may act as natural barriers compared to more open African plains caused a selection for smaller groups or gave way to human genetics that naturally selected for smaller groups. Of course, take it all with a grain of salt.
1. https://www.history.com/news/neanderthals-likely-lived-in-sm...
(Still lots of fun to think about. If I were to hazard a guess, it would be something along the lines of greater neoteny in h. Sapiens sapiens. For an analogy, compare dogs and wolves. Wolves have larger brains, but dogs have a greater ability to read social cues.)