Textbooks have always pictured the Neanderthal as a brutish and unintelligent being. The "experts" are going to have to rewrite textbooks.
Actually, those text books are from a hundred years ago. Since the 1950s Paleoanthropologists started seeing Neandertals in a new light.
They were still less intelligent than we were based on the conservative and unchanging tool set that they used versus our more sophisticated and constantly changing ones.
And yes, their brains were bigger than ours but not in the right places.
Neandertals had a sloping forehead above his thick brow ridges which meant he had less prefrontal lobes than us - critical to foresight and long range planning. We have a more domelike bubbly brain.
They also had less cerebral cortex than we which meant lesser general intelligence.
His brain volume exceeded ours because his skull was much longer than ours from front to back giving him an egg shaped skull compared to our bubbly one. That meant that they had more occipital lobes which are the visual center at the rear of the brain. This implies that they were more visually oriented than we, probably memorizing every nook and cranny of his territory.
They also grew up faster than us with a 10 year old Neandertal being the equivalent of a 15 year old Homo Sapiens. This required more growth hormone creating a higher metabolism which would have a greater impact on his personality. They were probably more quick tempered than us.
Simply a different species than ours.
As for the interbreeding between them and us I suspect that it was Cro-Magnon males who mated with Neandertal females. This was probably the result of violence between our two species where the victorious Homo Sapiens took captive brides. Sound familiar?