I also hate the phrase.
Yea boys and girls are different. But because people treat boys and girls differently, it's impossible to tell if they are acting differently because they are genetically pre disposed to it, or if they are acting differently because they are expected to.
I would guess, like most things, the true answer is somewhere in the middle. Boys probably have a bit more energy but when adults laugh indulgently when they run around screaming and fighting but frown and tut when the girls do, it's not a surprise that the differences are amplified.
From when kids are babies, girls are bought 'quiet' toys like dolls and dress up princess and bunnies and colouring sets and boys are bought toy drills and lego. People say 'oh he's handsome, he's going to be a heartbreaker' and 'shes pretty, you'll have to lock her up and keep her away from boys til shes 18'. Girls wear impractical clothes eg t bar shoes, dresses to school where they cant play as boisterously. It's a constant drip drip drip of girls are quiet and nice and boys are noisy and aggressive.
And that's what makes it so hard to fight against as if you brought it up, people can always argue that boys arent the same as girls and it's just a phrase. And on it's own it is just a tiny phrase but when it's part of a million phrases all saying the same thing then it cant not have an effect
And I worry that it can excuse bad behaviour. And I expect the people that say 'boys will be boys' are the same people who say 'what do you expect- hes a man' when someone complains that their husband doesnt ever get up with the kids in the night when they're ill, or 'men don't notice these things in the same way' when someone complains their male partner doesnt do any housework because they 'don't know how / dont realise it needs doing'. Like the mere fact they have a penis means they have completely different standards of acceptable behaviour, and certain tasks are beneath them.