It depends on the person. There are people who use both to mean people like us. While others say transgender is the umbrella term to mean anyone who is trans whether they are post-op or pre-op. Others use transexual to mean a transgender person who has undergone some form of medical transition.
Many people see the definitions as: Transgender: a blanket term involving anyone who doesn't idenitify as their birth sex, which involves non binaries. It can generally be used to refer to anyone regardless of whether or not they have taken hormones, gotten surgery, or don't want to medically transition. It seems to be more of an umbrella term. Transsexual: usually used to refer to people who don't identify as their birth sex and do go through medical transition. These people can still identify as transgender. I hope this was a clear enough explanation!
Transsexual is a bit more outdated and has a more negative connotation in general, but lots of people choose to use that term for themselves. Just don't use it on anyone without knowing that they're okay with it. I agree with the definitions of transgender that were given above. I wanted to add that I also frequently see transsexual being used by people whose definition of gender centers around physical dysphoria and/or intend to transition in the future.
