This is a common question ;-) There is absolutely no difference in meaning. Both forms are common in the spoken language, and even native speakers have no idea why they use one instead of the other. It has more to do with speaking quickly and with as few changes to the position of the mouth/tongue position as possible. If you are writing for business, science, or academia you should not use "got".

Of the three possibilities you mention: 1) I have to go 2) I've got to go 3) I got to go only the first two are considered to be 'correct' English. The first is standard English which you could use in any situation, formal or informal. The second is a little more informal, but still correct. It is appropriate for most situations, but should be avoided in more formal written situations such as academic essays or business letters. The third is something you might hear, but I would not advise you to use it. It is very colloquial, and not seen as correct grammatically. Sometimes you see the even more informal written form 'I gotta', especially in American English. Again, this is not something you should imitate. The only time you can use 'got' directly after the subject is when it is the past simple of the verb 'get' and the main verb of the sentence e.g. 'I went to the bank and I got some money' or 'I drank from a dirty glass and I got sick.' I hope that helps.