I once had a senior colleague tell me that Judith Butler had introduced the idea that sex and gender are different things. I had a hard time not laughing in this face. This is not Butler's idea, but one long predating her. Butler's ideas are much more radical, presenting the sexual, too, as socially constructed. Everyone already knew that gender is a construct.
This is not particularly complex, but I'll give it a try, making what is an essentially pre-Butlerian argument.
First, there are differences, like "men have penises and women don't." That's not a "gender" difference but a basic sexual dimorphism.
Next, there are secondary characteristics, like, men can (typically) can grow full beards and women can't. These differences are often difference in degree. For example, women have facial hair, but not as much of it.
Thirdly, there are differences having to do with bodies that are more gradated, like height and vocal pitch. To say men are taller is not exactly true. Rather, if you graphed out the difference between men and women in a given population, you would find that men are taller, on the average, but that any given man might be shorter than any given woman. In other words, men are not all between five nine and seven, and women are not all between four-eleven and five eight.
Now what gender does is to take whatever differences there may be, and run with them, imposing social meaning on them. It views all difference in behavior, dress, etc... as the natural consequence of a biological given. In other words, it confuses having a penis or a beard with the meanings that those things have. For example, I could have penis and carry around a woman's purse. There is nothing about this leather object, convenient for carrying around shit I might need throughout the day, that has anything at all to do with having, or not having, a penis.
In any given society, the treatment of the body varies by gender. You could imagine a society in which women and men have the same length of hair. A society in which only men, or only women, or both men and women, or neither sex, has tattoos on their body. But men have penises! Yes, we know that. That's not what we're talking about here.
Then we have social roles. They are organized around gender categories as well. They can be divided up differently in different cultures.
There are (at least) two extreme schools of thought that want to trace all gender categories back to biological sex. One is evolutionary psychology, one of the classic bullshit fields of all time. It is anti-feminist in its inclinations. The other is so-called "difference feminism," which to my mind is not feminism at all. What these tendencies do is take an overlapping chart of difference and make that difference essential and non-overlapping. So instead of seeing an alto and a tenor voice as pretty much two varieties of the human voice, one a little higher than the other, they divide all human voices into sopranos and basses. Since traditional gender roles assume the idea of non-overlapping behaviors, dress, and psychology, an ideological construction of these differences reinforces these gender roles even more, leading to a vicious cycle. After all, as long as people conform to these roles, it is easier to argue that they reflect innate differences.
The caricature is that feminists argue that there are no differences at all, or that men and women are the same. While some people might seem to believe this, nobody really does. Everyone recognizes that people have different genitalia and secondary sexual characteristics, that there is a difference in statistical distributions in things like height and vocal pitch. Everyone also recognizes that there are differences in traditional social roles. So nobody on the planet thinks men and women are "the same." All we ask is that we don't exaggerate differences in order to enforce stupid rules.
I've left out two things here. One is sexual attraction to the same or "opposite sex." The term "opposite" here is very loaded. The other is the body that is not easily classifiable in terms of sexual dimorphism.