Very generally I consider 'sex' biological, and 'gender' social, probably because of my background in Sociology. University likes having neat tearms for everything.

However, when I personally think of 'social' I don't think of 'as defined by society' but rather I think of 'as defined by one's personal view on how to conduct and see yourself in a social environment'. I also don't see gender (or sex for that matter) as a binary thing... in very broad terms I believe there is at least male, female, trans (a state of changing gender with a start and end point), neutral, and fluid (changing gender without an end point). Most likely many many more, or at least many nuances.

I don't think creating categories is inherently a bad thing. It's just not for everybody, which is what many people can't understand, and that is what creates problems so often.