I have known for as long as I can remember that sex and gender are different, and I have been guilty of going around thinking that most people know that, but they simply think that one must be kind to those who feel they were born in the wrong body.
Turns out they do not. A lot of people think sex and gender are the same, and that you can change sex if you try hard enough.
This came up because I was trying to find out if you can determine sex from blood, and looking at academic articles, many of which use the terms sex and gender interchangeably (declaring that you can determine a baby's gender from maternal blood, etc). Then I was talking to someone who hadn't really thought about it and explained to her the basics of why the terms are different. She was visibly shocked to discover that 'even if you take hormones???' you don't change your biological sex.
There's a huge mountain to climb still isn't there?