Regardless the grammar, don't you use "I gotta go" when you need to leave NOW whereas "I have to go" can include the sometime in the future?
I gotta go
This is an example of colloquial ellipsis, where words are omitted in informal speech. The complete sentence should be
I have got to go
According to the Cambridge Dictionary, have got to is just another way of saying have to, so this means exactly the same as
I have to go
"I gotta go" is much more informal.