If we break down the words, we should be able to differentiate between them: (The "trans" definition I copy and pasted, the other two were just improvised by me)
"Trans- (prefix): From the Latin meaning "across, over, or beyond." Medical terms containing "trans- " as a prefix include transcription, transfusion, transplant, transsexual, transurethral, transvaginal, transvestism, etc."
Gender: A mental identity in relation to masculinity, femininity, or lack thereof.
Sex: The biological traits characteristic of male, female or intersex.
Therefore, if we stick these together we get: (loosely)
Transgender: Someone whose gender identity is different than their physical sex. (Their gender is "across")
Transsexual: Someone whose sex is changed to match their gender identity. (Presumably through surgery)
So personally, I think of transgendered people as anyone who is gender variant. (Androgyne, Genderqueer, FtM, MtF, etc.) Transsexuals on the other hand are those who's bodies are changed through hormones and surgery to change their physical sex.