Sure. What, then, is all the current discussion about if not this fundamental question?Now, what is a man and what is a woman? It's not just a trick question, it's the most important part. We can change words to mean whatever we want them to, but if they're going to be people's identities, they need to mean something.
Let's start with sex. Sex is biological, and essentially constant throughout our species's existence. However, we have complex societies and sex has caused differentiation of people's societal status and roles, varying across times and different cultures. In that sense sex has been not just biological, but its consequences have passed into the sphere of the social. Is that biological definition adequate to explain this social component? Quick answer, no.
Hence, gender: a way to try to separate out the social differences caused by our different sexes, which "sex" is hopelessly unable to address.