The difference is how one uses the reasoning to apply to others.

Take freedom from discrimination. That isn't the argument used. Freedom from discrimination on the basis of sexual orientation is the argument used. While it works the same for those who were facing discrimination because of their sexual orientation, it doesn't protect those facing discrimination because of their gender identity, the conforming to gender norms, their sexual attraction, their age, their appearance, ect.

Now, the LGBT movement has been increasing what they want to include to be protected, but it has taken time and effort and even today you'll find some who support freedom from discrimination based on sexual orientation but do not support freedom from discrimination for gender identity. And in 10 years we will look back and other groups who were included which are currently not included (except for perhaps the most counter of counter cultures).

It also allows for some people to evolve a view of freedom from discrimination based on minority status. I think the better example of this is with freedom from discrimination based on gender minority, which has led to some people encouraging certain forms of discrimination if it is against the groups they think are okay to discriminate against.

For a very recent example of the legal impact of this, look at places that are currently passing laws to restrict the rights of adults based on them being 18-20 years old. Freedom from discrimination would cover discriminating against adults because of their age. But instead the law says that it is only freedom from discrimination based on being above a certain age and many people actively support discriminating and restrict the rights of adults because of their age being too young.

That is appealing to the concept of "freedom" to advocate for a specific goal, not advocating for the wider concept in general. Paraphrasing, what you've basically said is "movement based on [its members being not subject to] violence, [its members not being subject to] discrimination, and [ability to have the state facilitate its members] to marry. I agree that any reasonable person should agree with the first two, and at least cede the third under the current system - but specific non-prohibited actions have little to do with freedom itself!

The point is that this movement has been based around advocating for specific proclivities to be overtly accepted into wider society, rather than a general concept of freedom for everyone to live and let live.

If the only reason one is at odds with society is because of their sexual orientation, the two have the same effect. And for sure, positive-identity collective action is needed to organize to push back against the wrongs of anti-identity oppression.

But over time, these groups calcify and become prescriptive. For anyone that is otherwise marginalized, the two framings are drastically different. Which is why this identity politics then splinters into a million different group narratives - because it's ultimately based on emphasizing people's differences and tweaking top-down models in service of new "majorities" (quorums within subcultures, really).

I don't expect to get this across well, precisely for the same reason we're in the political bind we're in - most people can't fathom simply not meddling with others because they are extroverts and don't understand not being meddled with! But dictating other's personal choices is what each of these political teams has ultimately become fashioned around - essentially just bikeshedding over base bodily functions. From my perspective, whether you're insisting that I think a certain way due to the age old space ghost or a reformist political movement, you're still trying to force me into adopting an overprescriptive model.

(Likewise marijuana is being generally deillegalized, but the state continues on restricting the general freedom to alter one's consciousness. Getting to choose between specific state sanctioned options is not freedom!)