Specialization of labor in obtaining food different between Neanderthals and Humans

Is this specialization of labor one of the reasons that humans outcompeted Neanderthals?  The WSJ has this:

. . . In all hunter-gatherer societies there is a sharp difference between the foraging strategies of the two sexes. Men generally travel far in search of mobile prey that they need to bring down with well-aimed projectiles. Women generally go out in groups and search for good sources of roots, ripe berries or nuts, which they use their acute powers of observation to spot and collect. . . .
This sexual division of labor over foraging is not only far more marked in people than in most other animals (it was, arguably, the first "gain from trade" we stumbled upon, benefitting both sides), but it may be a relatively recent feature of our evolutionary history, invented in Africa just 150,000 to 300,000 years ago. Some archaeologists have concluded that Neanderthals did not practice it: that female Neanderthals were co-operative hunters with men, not gatherers. . . .