There is a popular notion that men and women are very different in their cognitive abilities. The evidence for this may be weaker than you expect. Janet Hyde advances what she calls the ‘gender similarities hypothesis‘, ‘which holds that males and females are similar on most, but not all, psychological variables’. In a 2016 review she states:
According to meta-analyses, however, among both children and adults, females perform equally to males on mathematics assessments. The gender difference in verbal skills is small and varies depending on the type of skill assessed (e.g., vocabulary, essay writing). The gender difference in 3D mental rotation shows a moderate advantage for males.
So from three celebrated examples of differences in ability only two actually show a moderate gender difference. Other abilities show no or negligible gender differences, Hyde concludes. Gender differences in ability may be overinflated in the popular imagination.
Worth noting is that the name of the game here isn’t to find gender differences in behaviour. That’s too easy. Women wear more make-up for example, men are more likely to wear trousers. The game is to find a measure which reflects some more fundamental aspect of mental capacity. Hence the focus on vocabulary size, mental rotation ability, maths ability and the like. These may be less subject to the vagaries of exactly what is expected of each gender, but that’s a shaky assumption. Indeed, it would be weird if different roles and expectations for men vs women didn’t produce different motivations and opportunities for practice of cognitive abilities such as these.
The real challenge is to find immutable gender differences, or to track differences in how abilities develop under different conditions. Without this evidence, we’re not going to be sure which gender differences are immutable, and which are contingent on the specific psychological history of particular men and particular women living in our particular societies.
One way of addressing this challenge is to look at how gender differences change across different socities, or across time as society changes. A 2014 study, ‘The changing face of cognitive gender differences in Europe‘ did just that, showing that less gender-restricted educational opportunities tended to decrease some gender differences but not others. In other words, increasing equality in educational attainment magnified some differences between the sexes.
You can read my take on this in this piece for The Conversation : Are women and men forever destined to think differently?
The Gender Similarities Hypothesis: Hyde, J. S. (2005). The gender similarities hypothesis. American psychologist, 60(6), 581-592
2016 update: Hyde, J. S. (2016). Sex and cognition: gender and cognitive functions. Current opinion in neurobiology, 38, 53-56.
Previously: Gender brain blogging: Sex differences in brain size, no male and female brain types.
Mind Hacks 
The claim is usually that the male brain is higher variance, which explains in an evolutionarily plausible way why males hold both most of the criminal records and Nobel prizes.
This appears to be true for several cognitive abilities but is also one of the most overstated gender differences I have ever seen. M/F variance ratios are empirically in the 1.10 range which would result in about a 4:1 ratio for out and out genius at 4 standard deviations from the mean. Even if you had to be this smart to win a Nobel prize (especially unlikely for literature and peace) the ratio of male to female Nobel laureates doesn’t really add up.
Even more of a problem with criminals. Those are less intelligent than your average person but I really doubt they are all retards – male : female differences in variance can’t explain male predominance here either
The idea that men and women differ in their variability has a long history, as well as a long history of refutation https://www.jstor.org/stable/3173639?seq=1#page_scan_tab_contents
The similarities/differences conversation also depends crucially on the way that psychologists frame their inquiries (among other things). The recent finding that reframing questions about spatial reasoning can do away with gender differences illustrates this nicely: Tarampi, M. R., Heydari, N., & Hegarty, M. (2016). A Tale of Two Types of Perspective Taking: Sex Differences in Spatial Ability. Psychological Science, 27(11), 1507–1516. https://doi.org/10.1177/0956797616667459
The cognitive differences are small but that doesn’t mean that the behavioral differences are small. Hyde did indeed find large sex differences in attitudes to casual sex and in physical aggression, two examples of sex differences that are cross-culturally robust and large. Also the sex difference in thing/people orientation is about 1 standard deviation and also a cross-culturally robust difference, the same goes for math or verbal tilt which is quite significant. In cognitive capacity men and women don’t seem to differ much, but more so in other aspects of behavior.