Although this a very old thread I want to respond because that has been my area of research for many years. Both terms belong to the same realm: Gender variance compared to the accepted binary order. However, transgender has been used as the umbrella to all gender variant people. Therefore, transsexuals are transgender. Transsexuals are individuals that feel in-congruence in their born body and demand corrective measures since their infant years. These corrections start by allowing the person to represent their congruent gender, to be considered and treated according to the represented gender and to address the physical "abnormalities" that create the in-congruence. Hormone therapy Replacement (HRT), surgical procedures to change genitalia and gender traits, such as facial feminization, body/face hair removal, breast implants, etc. Transsexuals ultimately aim to fully allow themselves to represent their congruent gender, including changing gender marker in their birth certificate, driver's license, SS card, passport, etc. It is known transsexuals experience Gender Dysphoria for as long as their congruency is not addressed.
Transgender people vary from occasional dressing in their opposite born gender, wearing female garments under their male ones, use of make up, etc, and there are individuals who are not transsexuals but opt to spend a portion of their life in their preferred gender, go out under that role and relate to others on the same page. The term Transgender has been abused through the last few decades by using it as a logical justification to transgress the acceptable binary gender in the mainstream. Transgenderism has existed since the times of Mesopotamia and literally there is no culture through history that does not have any evidence of such.
Politically and socially, transgenders are less supported than transsexuals because it is considered it is a matter of choice to them, contrary to transsexuals. However, academics have presented sound evidence regarding the innate validity to most cases of transgenderism, therefore the term 3rd gender/sex has been coined. This term covers all variances contrary to the exclusively accepted binary gender. In fact, many of this academics present a need to represent more than the two exclusive genders to understand the truths of gender and sexual preference. Equally, they have presented evidence regarding the validity of these cases as natural and innate through the 300 + species that show the exact same phenomena. However, homophobia has taken precedent in history to invalidate all, but the binary order and based on moral and religious beliefs.
Transgender and transsexuals, exclusively support a marked difference between them to the point that transsexuals can become offended to be referred as transgender. It works all the way around too.
In both cases, ignorance in this regard have created a series of conflicts of political and social order in which both, both realms are considered unfitted to society, as well, as a threat to society. Scholars attribute to this phenomena to sexual repression through history and superstition used by religious beliefs because they associate such - to times of natural disasters -that in their view, were created by the Wrath of God as punishment for sinful behavior. Such as the disasters occurred after Jesus crucifixion in the Jerusalem region.