A Transvestite is a person who wears the accoutrements of the opposite sex. Generally it is a sexual fetish although some people do it for other reasons. It may be an everyday thing or just something they do occasionally. It may be just one item of thing or a full outfit. They have no doubt that they are a man or woman. They are weird.
A Transexual is someone who thinks that all the problems in their lives are from being "born in the wrong body". They are mentally ill.