Transsexuality and transvestism (crossdressing) are different phenomena, although one person may have both characteristics to varying degrees.
Transsexuality refers to the innate, unshakeable conviction that you are in the wrong body: you really are a girl but somehow you have a boy's body - or vice versa. Transvestism, on the other hand, relates more to the desire to emulate the opposite sex, be it through dress, mannerism, role-playing and so on.
Curiously, transsexuals will often dress contrary to their anatomical sex, but do not consider themselves to be crossdressing.