"I got to ..." is a casual or substandard shortening of "I've got to ...".
"I have to ..." and "I have / I've got to ..." are synonymous in meaning. The version with "got" is more informal. I believe it is more common in British English than US English.
"have (got) to" tends to have more of a nuance of external obligation ("My boss told me that I have to work on Sunday"), whereas