Let’s not beat around the bush with complicated linguistic terms; the difference between “got” and “gotten” is relatively simple. First:
There is no such word as “gotten” in British English, and perhaps the only British expression containing the word is “ill-gotten”, which is an adjective meaning “obtained illegally or unfairly”.
If you learn American English, the situation is slightly more complicated. The past tense of “get” is “got”, just as in British English, but you should remember that:
For example, in the first case (receive, become):
I’ve gotten interested in chess. (= I’ve become interested in chess.)
And in the second case (have, have to):
I’ve got to go now. (= I must go now.)
Note that “have got” in the sense of “have”, “possess” is more common in British English and is often considered colloquial or even incorrect in American English. Also note that there are regional differences, and some Americans prefer “got” in the first case as well, but on average, the “gotten” form in the sense of “receive” and “become” is much more common than “got” in the US.