You're either sexually attracted to someone or you're not. If the only people you are sexually attracted to are people of the same sex, then congratulations, you're gay.
There are no situations in society where we segregate gay people from straight people. Gay people being gay have zero impact on other people's ability to live their lives. Some people might not like seeing a same sex couple kissing or holding hands in the street, or they might feel that the sanctity of the institution of marriage is impacted by allowing same sex couples to marry. But their own rights are not eroded in any way by gay people having equal rights. And that is all gay people have ever asked for: equal rights.
Now as for whether being trans is a choice or not, I'd say that's complicated. Part of the problem is that we are using "trans people" as an umbrella term to refer to, collectively, people like Jackie Green (who started identifying as a girl in early childhood, probably thanks to a homophobic father who thought letting his son play with dolls would make him gay), autistic teenage girls who showed no signs of gender distress until adolescence before suddenly expressing a desire to opt out of womanhood, heterosexual men who suddenly decide they are women in middle age having married and fathered children, people like Eddie Izzard (who used to just consider himself a cross dresser) and people like Karen White and Isla Bryson (hopefully no explanation necessary). Clearly the reasons why these different groups of people are identifying as trans are not the same. And once again, it's the teenage girl cohort who are most damaged by treating them as the same, because pretty much all the medical research on paediatric transitioning was conducted on male subjects. Either way, I think it's clear that some of these people are in genuine distress (and the appropriate form of care may or may not be medically transitioning) but others are not and have effectively "chosen" this path.
Regardless of whether it is "chosen" or not, the main difference between gay people and trans people is that the rights sought by trans people are rights no other group has (namely the right to access single sex spaces and sports for the opposite sex) and which have a directly harmful impact on women and other minority groups with characteristics they definitely have not chosen.
If using single sex spaces corresponding to their own biological sex really is unthinkable for trans people, they need to be campaigning for their own spaces. And if being referred to by their biological sex is so distressing, they need to come up with their own vocabulary to describe what they are, rather than appropriating the only words we have to refer to the opposite sex and leaving the opposite sex without the language to clearly identify themselves.
And of course, the "trans rights" issue also has a specific impact on gay people. Not just lesbian dating apps being inundated with male users, but also children being encouraged to transition when the best evidence we have suggests that if left well alone they would grow up to be perfectly normal gay or lesbian adults, with all their body parts intact.
If you made it to the end of this post and still think there's no difference between gay people and trans people, I'm not sure what more I can say to convince you.