if you use "gender" you are talking about being woman or man in every sense (as in gender studies, gendered patterns of behaviour, etc). if you use "sex" it has more to do with specific body functions or morphology. the difference is pretty good made in academic millieus. the doctors speak about sexual disfuntions, for instance, but not gender disfuntions or about sex transmitted diseases and not gender transmitted diseases. of course, you don't have to expect everybody knows the difference as it's been for a long time normal to speak of "sex" with a very wide meaning.

Gender has one meaning, male or female. Sex has two meanings, male or female, and the other meaning I don't have to say. Gender is also a nicer word to use.

There is the technical answer, and the commonly-understood answer, and they differ, which makes it hard for the language learner. Technically, “gender” refers to the socially acceptable ways of behaving. In the US, this is commonly masculine (manly) and feminine (womanly). In some other cultures, there are more than 2 options. This is different than “sex” which refers to the physical arrangement of one’s body. In the US, this is commonly either male (a man) or female (a woman). People born with intersex genitalia are sometimes sexed at birth, surgically, to choose one or the other. But in common usage, these differences are not widely known. Also, since the word “sex” that means the physical arrangement of the body is the same word as “sex” that refers to the sex act, this is a word that is not commonly used when you’re trying to be polite, such as on a government form. So what the form is usually asking is your sex, but they put the label “gender” on it because they are trying not to use the word “sex.” But for a non-native speaker, these differences are usually hard to explain, especially when common usage is not the same with academic correctness. I hope this helps clarify.

Gender is a role. It may include society's ideas of how you are supposed to act, dress, etc. Sex means whether you have boy parts or girl parts. It's a physical thing.