A big misconception is that women are wired to be more "caring", "soft", and less competitive than men. This is a product of socialization more than biology, little girls are raised to adhere to their society's idea of feminine. For example, I don't wear makeup and do my hair because I like to or because I'm more vain than the average man, it's just what women are supposed to do. Women are also extremely competitive, men just don't notice. Believe me when I tell you 90% of the reason women put on makeup and dress up is to trump the competition moreso than attract men.
Prove it. I can cite several chemical / physical differences between men and women, and how these differences can create differences in behavior / personality / ability / whatever. To say that it's either the environment or the genetics is a pretty extreme statement. Both society and biology play a role.
Did you know neanderthals had larger brains than the average modern human? I would hardly call them more intelligent.
The reason men have larger brains is because they are larger and have more muscle mass than women, more neurons are needed for motor control and thus the mass of the brain is larger.
Relative to the size of said Neanderthals, their brains are actually smaller, which is a much more useful comparison.