Increased blood flow in areas of the brain in women

Not really, no. Brains develop along with experiences, so two identical brains that go through different lives will develop differently. Someone who lives in solitary confinement for 10 years will have dramatically different brain activity than someone who doesn't, for example.

So, sex differences in brain activity could be genetic, could be cultural. No real way to tell without putting babies in experimentally controlled developmental trajectories, with double blind caretakers who are selected for androgyny.

And as for the "well, but this is still evidence there are two brains, regardless of cause" argument: Not necessarily. There are lots of axes on which we can slice populations into two statistically significant groups. In order to show that there are two sex brains, you'd have to show that the sex differences are more significant than other partitions.

Plea to other users: Please don't downvote or flag my parent poster. They are politely citing a commonly believed theory about sex that merits discussion.