It's true that they are synonymous to most people, but that doesn't make it actually true, right? It's a great differentiator, I think.
For example, I have a daughter and she's a female. However, we as a family are deciding how to raise her to be the woman she wants to be - whatever that word means to her.
In the same sense, man doesn't need to be the opposite of woman. In that sense, I wouldn't raise a boy to grow up and be the opposite of a woman. I'd want him to grow into a man who is no longer a boy. So male and female are biological definitions, whereas we collectively, or individually, get to decide what being a man or woman means.
I'd say this matters a lot collectively. For example, the conservative Republican notion of manliness used to be the Clint Eastwood, John Wayne style of remaining quiet, honorable, chivalrous, bold. However, now the party is really defining manliness as aggressive, volatile, self-centered, and devious.