Where did the idea that sex and gender are different come from?

Correct me if I'm wrong, but I don't think the idea can be attributed to a single group, less a single individual.
There have always been cultures who believed there was something like a third gender, that sometimes meant intersex people, but also trans people. In some cultures those people were held in high regard, as they were believed to be of a very spiritual nature. (To put it very simply, I can't really think of the correct English terminology right now.)

For example:"Texts suggest that third sex individuals were well known in premodern India and included male-bodied or female-bodied people as well as intersexuals, and that they can often be recognised from childhood." (from the "third gender" entry on Wikipedia)

If you're specifically asking when the notion first started occurring in more modern science, I can't really help you. Sorry. ^^

 

According to Wikipedia, in Western science it's 60s-70s, and 70s are mentioned as the time when it became a thing in the US.

Also, remember it's something that gets lost in translation. In my language, Polish, there's only "sex" and "sex identity" would be a substitute for gender. Sometimes they also say "social sex" to refer to gender in the modern sense.

In general, in Wester world such ideas were sparked by psychoanalitic theory and its feminist applications in the early 20th century, and became more of a science in the 2nd half.

 

well, you can try reading this https://en.wikipedia.org/wiki/Sex_and_gender_distinction

As for Polish, you mean that we have no word for "gender"? We just never did. We have something that describes grammatical gender but it's a word closer in meaning to "a kind". It's just that we have no word and need workarounds, because the concept of gender vs sex as different things was coined by English speakers and we had to find our own way to express it. So, we use descriptory workarounds. "Sex identity" is most popular and accepted.

 

It's actually better to ask "when did they start thinking of them as the same thing?" imo! They changed back in the 60's-70's in Western society, okay, but when did they start? Betcha it was around the beginning of the Industrial Revolution! That's when they invented the word 'heterosexual'