@brianjakub,
I think we clearly outwitted neanderthal in practice. In potential, nobody can tell. But in practice Cro Magnon came with a better technological package and relatively quickly thereafter neanderthal disappeared. That can't be a coincidence. The two species competed and our ancesters won.
The traces of neanderthal genome left in us predate the European colonisation by sapiens, they go back to 60,000-80,000 years ago when the two species cohabited in the Levant. Interestingly, they would result overwhelmingly from male neander x female sapiens, rather than vice versa.