This New Yorker article gives a nice blend of personal, scientific, and species history. It would be interesting to know if what we picked up from our bigger brained but ultimately less successful cousins was just random or if it includes anything useful.
And as to the question of how “we” won, I am still inclined to speculate that there’s some kind of tie in between our capacity to learn from and cooperate with peers, a willingness to take direction from others (sometimes “blindly” so), and the tendency towards art and decoration – especially that of our own bodies to help identify “team” members in conflict situations. But now it looks as if the competitive team spirit and blind obedience to authority that won us the world may be about to become our undoing. It would be ironic if some admixture from the Neanderthals were to be what we need in order not to destroy ourselves in religious conflict.