Lest we forget - Neanderthal’s brains were larger that H Sapiens. i.e., they were smarter. Another fact which hasn’t filtered out to the general public is that the size of our brains are decreasing over time. i.e., we are getting dumber.
On the other hand it's a very well made observation.
The cold hard facts are the Neanderthals were working on becoming MODERN HUMANS as in Sapiens Sapiens more than 300,000 years ago. The current crowd who imagine themselves to be MODERN HUMANS started only 180,000 years ago.
That's a full Glacial Cycle ~ the kind of experience that imbeds itself in the genes. The more recent ability to outbreed our range does not really make us advanced ~ just more like bunny rabbits.
The common assumption is that Neanderthals were stupid. Stupid people don't survive in cold climates. Mistakes are more likely to result in your death.