In one recent study, Tim Kaiser, Marco Del Giudice, and Tom Booth analyzed personality data from 31,637 people across a number of English-speaking countries. The size of global sex differences was D = 2.10 (it was D = 2.06 for just the United States). To put this number in context, a D= 2.10 means a classification accuracy of 85%. In other words, their data suggests that the probability that a randomly picked individual will be correctly classified as male or female based on knowledge of their global personality profile is 85%.
Consistent with prior research, the researchers found that the following traits are most exaggerated among females when considered separately from the rest of the gestalt: sensitivity, tender-mindedness, warmth, anxiety, appreciation of beauty, and openness to change. For males, the most exaggerated traits were emotional stability, assertiveness/dominance, dutifulness, conservatism, and conformity to social hierarchy and traditional structure.And the differences are biggest in those nations with the most gender equality: the Scandinavian countries always show up as having the biggest difference, followed by the Anglosphere, with conservative Muslim nations at the bottom.
For instance, if you look at the density distribution for agreeableness, the average difference between males and females is only about .4 of a standard deviation. However, if you look closely you can see that there are way more women than men who are super-agreeable and way more men than women who are super-disagreeable. It's likely that the behaviors carried out by those tails have a huge impact on society.So men are on average only a little more obnoxious than women, but at the far end 90 percent of the raging assholes are men.
None of this says anything about what causes the differences; the fact that the magnitude of the difference varies between nation implies that it is partly cultural. But on the other that the fact that the difference exist everywhere implies that they are partly biological as well.
As to what, if anything, we should do about this, I would say that this is one of the major long-term questions we need to somehow come to terms with.