It's one of the classic nature vs nurture questions - are men and women different due to societal pressures or are there genetic/hormonal factors that allow differences to be observed? The documentary does a good job of presenting evidence from both sides and rather than give us an answer leaves us to judge. The fact the show can't decide whether there's such a thing as a male or female brain renders the title question a bit moot. So you'll be disappointed if you are hoping to get an answer to that. The rest of the show is very interesting though and the parts that stood out for me were;
Showing that monkeys have a gender bias to toys that is the same as in humans. Not sure monkey parents could introduce that bias (although not completely impossible).
Showing that research suggests a correlation between testosterone levels in the womb and the severity of autism. It was also interesting that autism could be seen as extreme "male" behaviour. Women are also exposed to testosterone so can still be autistic but this result also fits the observation that autism is more common in males.
That some of the observed differences in spatial awareness disappear when tested in a way that the subject doesn't see it as a spatial awareness test. This raises two important issues - the tests may have a bias towards a gender and that being told "women can't do X and men can't do Y" can be a self fulfilling prophecy.
That medical treatments may become gender specific. I suspect this one may be skipped over by personalised medicine which is already on the horizon. I guess gender specific medicine could still be a useful stepping stone though.
The one aspect of the show I didn't really enjoy was the male and female presenter format, I can see why there was the temptation to go for that angle but it felt a bit forced having them fall on different sides of the fence. Was that supposed to be a male vs female bias? I also felt one of the presenters was cherry picking the evidence eg refused to accept genetic components for things the other gender is apparently better at but apparently being ok with it when it was their own gender with an advantage. Or maybe that's my gender bias? My sample number of 2 suggests it wasn't.
Usually the answer with these polar explanations is the obvious "a bit of both" and I definitely think that's the case here. With regards to nature we'd be foolish to think the X and Y chromosomes are solely responsible for behaviour, For example if having a Y chromosome is " - 5" for having empathy, there's probably at least "+ 10" for empathy floating around on the other 22 chromosome pairs. this would explain the variety within a gender as well as the fact there are crossovers. Just as personalised medicine will supercede gender-specific medicine, so too will the genome overcome the X and Y chromosomes with regard to behaviour.
In terms of nature; I do think society needs to make a concerted effort to iron out gender stereotypes as the evidence presented in this episode shows how it can make the brain behave accordingly. Far better to convince both genders that they can be ambitious, capable in any discipline and have the full range of emotions than to put either one at a needless disadvantage.