It's very simple. "I've got to" is the present perfect tense, whilst "I got to" is the past simple tense. We use them in different ways. "I've got to go to school" - I must go to school. "I got to go to school" - I was lucky, I had the opportunity to go to school when I was a child, but many children around the world didn't have that opportunity.

I'm guessing that your question is this: What is the difference between: It's late. I've got to go home. It's late. I got to go home. The answer is simple - the first is correct, informal spoken English. The second is lazy, incorrect English. In fact, both are versions of the construction: It's late. I have to go home. (Tenho que ir para casa - just like Portuguese) The verb 'have' is quite a weak sound. As you know, it tends to contract to 've. 'I've to go' doesn't sound good, so we added the word 'got' to make it sound more complete - 'I've got to go'. Then the "ve" disappeared completely, and we are left with 'I got to go' (sometimes - even more incorrectly - written as '"I gotta go"). So in fact we have four versions of the same expression: 1) I have to go. 2) I've got to go. 3) I got to go 4) I gotta go. (1) is the most correct/formal. (2) is fine for speaking and informal writing (3) and (4) are ok to say, but don't write them. I hope that answers your question.

You could be more expansive in your knowledge of English. For example, rather than force yourself to use a phrase with "got," you could shift entirely away from such awkward phrasing and use something like this. "I need to __________ ____________ ___________."