I don't think it's so much purely that differences are based on individual personality, more that our society has a highly gendered way of allocating qualities as 'male' or 'female', and they tend to 'tidy' ungendered behaviours like energetic, running-around play into a 'typically male' mental box.
In your update, what strikes me is that other people keep telling you how different it is to have girls and boys, when what they're doing is extrapolating general principles from their experiences of having one or two individual children of either sex, and from their own gender expectations.
My two SILs (who have two boys and one girl each, aged from their late teens to 30) are always telling me similar. Obviously, I don't have a daughter, so have no personal experience of bringing up a girl, but it's obvious to me that what they perceive to be innate differences in their sons and daughters are at least in part down to their very different expectations for their girls.
In both families, the girls are the youngest by several years, and both were the result of their mothers' passionate desire for a daughter 'to do girly things with'. It doesn't seem entirely coincidental that both girls are indulged compared to their older brothers, are obsessed with their appearances and clothes, and have done conspicuously badly academically compared to the four boys. I don't think this means that girls are shallower, less ambitious, clever or independent-minded than boys in general, only that these girls are the result of a certain set of parental expectations about what it means to be a girl and a youngest child.
God, that got long.