There doesn't have to be a behavioral difference between men and women for societies to trend towards patriarchy. The simple fact that women bear children and are pregnant for nine months is enough to have limited their political prospects in pre-modern times.
A society of antiquity that allows women to fight in battle will have fewer women to bear children, which means fewer bodies to grind in the next war. Societies that forced their women into domestic roles could have faster manpower replenishment and win more wars, taking more territory and spreading their customs. Political power was inexorably tied to warfare, and women, often lacking the crucial military influence enjoyed by generals were locked out of the elite decision making classes.
Today, we no longer fight wars of annihilation, and a society that promotes women in the workforce has access to more brilliant minds than one that doesn't. I think there's more to gender relations than human psychology. Natural selection among human societies can be a powerful driver of cultural change.