Question: is there a significant difference between the fight for Gay Rights and the fight for Transgender rights?
Yes.
The fight for gay rights sought to provide equal treatment under the law to any other person. It was largely a campaign to 1) decriminalize homosexual intercourse and 2) allow same-sex couples to have access to the same treatment in law as opposite-sex couples.
The second included not just the right to marriage in the casual sense, but the right to access tax breaks afforded to households, the right to be next of kin for one's legal partner, the right to inherit automatically without needing to be specified in a will, the right to make medical decisions on behalf of an incapacitated partner, and the right to obtain common-law spousal protections for long-term partnerships without a formal marriage.
Those are things that any opposite sex couple have a right to, and which were definitively denied to homosexual people.
Nothing that was sought as part of the campaign for gay rights imposed an obligation on other people, nor did it in any way interfere with any other person's rights.
The same is not the case for transgender rights, generally speaking.
First off, let's be clear: There are some protections that I think transgender people should absolutely have. They should not be denied employment, housing, or access to public services on the basis of how they present.
The problem is that the things that are being demanded under the guise of rights are not rights in the first place, nor are they in any way equal. What is most often being demanded are special privileges - and ones that are obtained by directly reducing the rights of others.
Almost all of the "rights" being demanded require that a person's claimed gender identity be granted primacy over other people's material sex. Thus, the demand that transgender people be granted the privilege of using toilets, showers, changing rooms, and prisons on the basis of their said-out-loud gender identity where for everyone else use is based on the material reality of their sexed bodies. This is, in effect, granting one group the right to violate other people's rights
on the basis of their personal beliefs.
When we look at the impact of these demands, one thing we see is that there is an extremely disparate impact. Granting transgender identified females the right to override sex-based policies and grant access to male-only spaces presents no material risk to the males who use those spaces. It does not increase the risk of males being exploited, abused, harassed, or assaulted by females who present as "men". The rate of sexual offenses committed by females against males is extraordinarily low.
On the other hand, granting transgender identified males the right to override sex-based policies and grant access to female-only spaces DOES present a material risk to the females who use those spaces. The rate of sexual offenses committed by males against females is very high - and there is existing evidence that transgender identified males retain male patterns of criminality and sexual offenses. There is no evidence to support the narrative that transgender identified males are no threat to females.