Short answer
Longer answer
Firstly, it is important to start with some disclaimers as the biology behind the entire spectrum of both gender and sex is a complicated topic and is beyond the scope of this answer. We will, however, get into this in the near future! To answer this question, we will therefore only be talking about cis-men (people who were assigned male at birth, and identify as male) and cis-women (people who were assigned female at birth and identify as female).
Secondly, we have to be careful in our interpretation, because throughout history the answer to this question played an important role in the debate of gender equality. For example, brain science results showing that men and women may differ, in any type of way, are taken as the reason for stereotypical gender differences, or even as support for those claiming that women are less intelligent than men. To be clear: This is not the case! Men and women were found to be equally intelligent. To cut a long story short: the male and female brain do differ, but they are more similar than they are different. Moreover, the differences between male and female brains are smaller than the differences between male brains or the differences between female brains.
Thirdly, even if we find a difference between male and female behaviour or skills, it is very difficult to say whether this difference is innate (due to genetics/biology), or due to for instance stereotypes in culture or upbringing. There is for instance a complex interaction between heritable and environmental effects on scores on IQ tests.
Now that these important points are settled, let us look at where there actually may be some differences between the brains of men and women.
We can often see that women and men differ based on their physique, this is due to their genetics and biological development. The genetic data that we receive from our parents determine whether we are (in most cases) born as a boy or a girl, and what types of hormones (most) stimulate our growth. Men will have more of the typical male hormones (for instance testosterone) running through their body, and women will have more of the hormone oestrogen (Check the links below to find out more about these hormones).
These hormones have specific roles and the level or intake of more of these hormones may actually boost masculine or feminine bodily features, as well as certain skills. That is why, for instance, the Olympic committee has argued that women with a testosterone level higher than a set maximum, are not allowed to compete with the other women, as it would give them an unfair physical advantage. A typical effect of high levels of testosterone is that it boosts your confidence. As an example, women with chronic anxiety have received short-lasting testosterone treatment, which much improved their symptoms (i.e. made them more confident). Importantly, this does not mean that all women with anxiety should be treated with testosterone because it has considerable side effects that may not be desirable (like stimulated hair-growth, or other behavioural side-effects). Still, it shows the impact of this hormone on self-confidence and anxiety. Interestingly, multiple studies have shown that men with on average higher testosterone levels tend to overestimate their own abilities a little, while women as a group tend to underestimate themselves. So, in the bigger picture, there appear indeed to be hormone-induced differences in behaviour between the sexes.
So, are there also differences between the brains of men and women? Male and female brains are very similar. They have the same shape and the same brain regions. They do differ though, as women have smaller brains, and fewer brain cells, even if we take into account that they tend to have smaller bodies. In the first place, as stated before, this does not mean that they are less intelligent! A study looking into how these brain cells communicate with each other found that brain cells in women seem to make more connections with other brain cells than those of men, and that women use the energy sources of their brain cells more efficiently. This may be why women need fewer brain cells, to be equally intelligent.
To conclude, men and women do have slightly different bodies and brains. They are, however, more similar than they are different, and are equally intelligent.
Further Reading (Dutch)
Het VrouwenBrein, Iris Sommer
Dit doet testosteron écht met je gedrag – De Correspondent
Further Reading (English)
Sex Differences in the Brain (brainfacts.org)
https://en.wikipedia.org/wiki/Sex_differences_in_intelligence
Testosterone & Oestrogen:
Testosterone — What It Does And Doesn’t Do – Harvard Health
Oestrogen’s Effects on the Female Body | Johns Hopkins Medicine
Hammes, S. R., & Levin, E. R. (2019). Impact of oestrogen in males and androgens in females. The Journal of clinical investigation, 129(5), 1818–1826. https://doi.org/10.1172/JCI125755