There's not really any difference. I was surprised to see, when I taught in Madrid, that many younger students are being taught "to have got" instead of "to have" as far as ownership or possession is concerned.
Even though we use "have got" a lot in BrE, I never considered it a verb in its own right. I would say "I've got a very cute cat" or "I've got a silver car" in my own day-to-day speech but I only ever taught "I have a very cute cat" and "I have a silver car".
In all the time I was teaching in Madrid (ever) I taught only (nothing else) the verb "to have", not "to have got". Obviously, I taught other things but in terms of verbs to do with ownership and possession, I taught "to have" and nothing else.
Yes, as it was mentioned previously by emsr2d2, have got and has got which show possessions are both typical in British English, however in American English have and has are preferable. By the way I haven't heard of had got.To refer to past you make use of had, both in American and in British English.