They technically are the same, but have to do is usually used more since the "got to" isn't required. "Have got to" usually means a stronger version so to speak. The main difference is that "Have got to" is not used when it is negative.
For example, it could be used with "I have got to go," but when it is said that "I haven't got to go" then it would be incorrect. If it is negative, it would be "I don't have to go."
I hope I helped! If you have any other questions, feel free to contact me~