Firstly, this is completely orthogonal. But it's also improper reasoning.

If Neanderthal had bigger brains (they did) or had different cognitive abilities, there's a chance they were baseline smarter than homo sapiens at the time.

Being perhaps a little smarter doesn't mean you win the evolutionary game. There are so many factors at play.

Hmm, more smarter? https://en.wikipedia.org/wiki/Brain_size#Cranial_capacity

Not the lady Neanderthals:

> average Neanderthal cranial capacity for females was 1300 cm3 and 1600 cm3 for males. [Modern humans, 1473 cm3.]

Nor the dude Neanderthals, since they were using the swollen brainparts for vision and coordination:

> Neanderthals had larger eyes and bodies relative to their height [...] when these areas were adjusted to match anatomically modern human proportions it was found Neanderthals had brains 15-22% smaller than in anatomically-modern humans.

Edit since I don't even agree with the concept: even if the extra capacity was differently distributed such that they had more ... powerful? ... executive functions, what's smartness? More imagination, OK, more self-restraint, more planning. More navel-gazing, more doubt, more ennui.

Or it could be more communication, often proposed as what gave sapiens the edge. Chattering bipeds. It's an association between the brain doing something and the species proliferating, that's what we're calling smart, but doing what? It could just mean our ancestors were compulsively busy. Same thing as smart, perhaps.

> Being perhaps a little smarter doesn't mean you win the evolutionary game. There are so many factors at play.

Considering most human groups have a % of Neanderthal DNA, they didn't exactly lose... Based on the % of Neanderthal vs. Sapien DNA, it seems Neanderthals were simply outnumbered.