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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?

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    More context is needed to determine that. "I gotta go" is more informal and perhaps a bit rude, though it does have some implication of immediate action.
    – user3169
    Sep 12 at 5:20
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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.

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