This is a subject that is confusing for many. When it comes to genetic males presenting as females there are really three distinct groups, and their aims are very different:
Crossdressers are usually men who do not identify as women, are often straight and crossdressing for release and often a sexual charge from the clothing and the exhibitionism. There’s some disagreement about whether crossdressers should be under the rubric of being transgender. Some crossdressers will progress to being transsexuals, and I suspect that there has to be some transgender aspect present internally. The older term transvestite is now considered pejorative by the crossdressing community.
Drag queens are gay men who dress as very exaggerated females, usually for the purpose of performance. As gay men they still identify as being male and not being interested in being women.
Transsexuals are men who identify internally as female and are primarily interested in resolving their gender dysphoria by living as women. This can include a number of steps. Dressing in normal women’s clothing and facial hair removal would be the minimum. Hormone therapy is usually needed. Reassignment surgery is not always pursued for a number of different reasons.
The major aim of a transsexual is to live as a woman without being hassled. Because of this, “passing” or appearing indistinguishable from a genetic female is very important to most transsexuals. Self image is another reason why many transsexuals find this important.
The above is based both on my direct experience as well as what I’ve read. I’m not claiming these are entirely closed boxes. It is perfectly possible for someone to be a drag queen, and also an effeminate gay man who is internally trans and finally finds he needs to transition. It isn’t the case that these distinctions are more than a convenient shorthand.