"I have to go" shows hat you have previously committed to be in another place, at a specific time. "I gotta go" is way more urgent (to American English speakers) than "I have to go". The difference is more of a personal, emotional commitment in "I gotta go" than "I have to go". "I gotta go" is used in situations like, "I gotta go now", my wife is having our baby (in the hospital). Whereas, "I have to go", is used in situations like, "I have to go" to class at 1:00pm today. "I gotta go" is closer to "Life and death" than "I have to go" (someplace).

"Gotta go" is like a slang contraction of "have got to go". English speakers are a bit lazy and like to cut out words when the context is obvious. If you say "I've got to go" really quickly, it soon starts to sound like "gotta go"

'I gotta go' means 'I have got to go' (more informal) and "I have got to go'. Both mean the same.

"Gotta go" is slang.