A post titled Neanderthals had differently organized brains reports evidence that Neanderthal brains did not have a larger volume devoted to intelligence (or at least that part of intelligence needed to handle social interactions in large groups) than humans.
A key fact is that “eye-socket size is correlated with latitude” – at least within a species.
Neanderthals were adapted to high latitudes, and had larger eye-sockets.
That suggests a relatively large part of their brain was devoted to the visual cortex, and it seems somewhat plausible to suspect that much of that involved low-level processing needed to make up for darker conditions at higher latitudes.
So Neanderthals’ larger skull size doesn’t imply any important advantage.