It is to do with the difference between sex and gender.
Sex is easy to define - its whether you are born bodily female or male (not how you identify)
Gender is disputed, some people say it is social roles forced on men and women (like women are delicate, caring and men are unemotional and strong etc) . Some people say they dont have a gender.
Gender critical feminists are critical of all gender roles forced on their respective sexes and think that gender roles and norms are harmful to both men and women.
Transsexuals say they want to be the other sex - as in change bodies, transgender people want to change gender.
Obviously, as gender is difficult to define some transgender people say they don't need to change anything about their body because its their internal gender they are changing - look at someone like Danielle Muscato or Gregor Murray (
https://www.pinknews.co.uk/2019/08/13/scotlands-transgender-politician-gregor-murray-isolating/) who are born male and say they are women, but have not made any attempt to assimilate.
FYI I do not think these people are women.
confusing the matter further is the fact that some people/orgs use the word gender when they mean sex.
There's also the age difference of the terms - transsexual is far more likely to be used by someone of the older generation like Buck Angel (a transsexual former porn star). Transgender is more likely to be used by younger people.
Hope this clears some stuff up.