From:  http://discovermagazine.com/2011/evolution/23-how-we-won-the-hominid-wars/article_view?b_start:int=1&-C=

On Neanderthals:

 The difference is that these early populations of our species had developed the ability to invent new tools, like sewing needles that were useful in producing warm, body-hugging clothing. Preserved beads and stones suggest that they, but not the Neanderthals, maintained social networks over vast areas. 

I add:

Of the advantages of our species over the Neanderthals, I suggest social networks were the most the most important.  While it is clear that the advantages of our inventiveness and our ability to adapt was critical for success, I believe that our use of social networks gave us the option of mutually supporting each other, even over great distances.

They permitted us to share those new inventions and adaptations.  In this manner, developments which would have assisted a small group of humans to survive spread out rapidly and gave the advantage to human beings in a given regional area. Eventually, the knowledge and skills would spread to all human beings, anywhere on the continent and beyond. 

It seems to me that without the benefit of this sharing, whatever advantages had been gained by the innovation would be limited only to a very small group of humans. A group of perhaps 15 or 20 humans could not effectivelyrealize the benefits in order to compete against a population of many thousands of Neanderthals over a large regional area. 

In other words, it is the  opinion of this retired educator that education was the tool which gave our species victory in the competition for control over the  resources on this planet.