What, L, G, B and T? All at once?
As discussed elsewhere under this heading, it has become customary to talk about gay, lesbian, bisexual and trans questions all under one umbrella (to which may be added queer, intersex, allies and probably half a dozen other somewhat-overlapping labels). From my perspective there is too much in common between the parallel struggles for acceptance of the non-heterosexual and the non-cisgendered for the two to be neatly separated - and if they can't be neatly separated, I question whether it is worth separating them messily.
As DonJaime acknowledges (see above), the struggle for gay rights and acceptance is in some senses inevitably bound up with the wider fight for acceptance of differing gender roles. It's not just about who can have sex or fall in love with whom; many gays might otherwise conform closely with established gender roles, but many don't. Some gay men are naturally uncomfortable with the expectation that they will be effeminate, or particularly concerned with their physical appearance - and many lesbians are not remotely butch. On the other hand, many do in fact wish to be accepted while possessing traits that are contrary to society's expectations of their sex and gender.
So fighting for acceptance only for homosexuals who otherwise entirely fit the pattern that mainstream society has assigned to their sex would mean fighting for acceptance just for some homosexuals; it would be like fighting racism while insisting that it's acceptable to discriminate against people based on characteristics that happen to be heavily correlated with race in your society. What this means is that the LGB fight is, to a significant extent, inherently also a T fight, where T is interpreted to mean transgendered, not transsexual - a distinction which is important, but far from absolute (some prefer to just say 'trans' rather than emphasising the distinction).
The transgendered are particularly concerned with not conforming to the gender roles assigned by society to people of their sex; transsexuals are specifically uncomfortable with the sex assigned by biology. There is, needless to say, a great deal of overlap between the two - but also enough dissimilarity to cause considerable friction at times.
If we're already fighting for the acceptance of people who do not conform to traditional gender roles, where does that leave us with respect to transsexuals? It is true that some transsexuals are very keen to conform as completely as possible to the gender role usually assigned to the sex they were not born into - and that some are also very good at achieving this. But once again, many are not! The fight for acceptance of transsexuals cannot be only the fight for people to be able to change their sex outright and be accepted as whatever sex they choose; it, too, needs to take in its stride the existence of many people who just do not conform very closely to society's gender norms, whether or not they want to.
That said, there is considerable non-overlap. Yes, transsexuals generally suffer from gender dysphoria, which is largely irrelevant to homosexuals, who also will seldom have to consider the ramifications of changing their identity, appearance and probably their name at some point in the middle of their life. Yes, the most fundamental concern of homosexuals is the right to sleep with or love whoever they like, which is largely irrelevant to transsexuals in most countries, although the right to marry is a shared concern.
There are of course practical, not just theoretical concerns with seeing something as part of a wider struggle. What if you disagree on a quite fundamental level with some of the things people are saying in the wider movement? What if the voices speaking out for your particular concerns are lost in the hubbub? These are questions that seemingly affect every movement everywhere, which is why I won't speak to you if you identify with the People's Front of Judea rather than the Judean People's Front. You splitter. The truth is there are no simple and correct answers here, which is not surprising when you consider the complementary questions - what about all the fundamental agreements? What if your voice is heard because you're part of a wider movement?
There is a dream that some people have - a utopian ideal - of a world where people are free to explore self-expression and lives outside of the mould of traditional gender roles and relations. It's a big part of what feminism is about, and it's a lot of what gay liberation is about. It is a big chunk of what trans people need, too, along with some consideration for people who feel for whatever reason that they can't be happy with the sex of the body they were born in. From where I'm standing the effort to keep the struggles separate would cost a good deal more than it would gain - but then, it is true that I'm not aware of any fundamental disagreements I have with most of the people involved on any side, and I suppose I might feel differently if I did. However you look at it, though, there is enough in common between the challenges faced by these separate but overlapping groups that they can't help but fight many of the same fights. That they also need to fight a number of independent fights strike me as a lousy reason not to stand in solidarity.