This is a very interesting question in my opinion.
Well, I will try to answer it
One difference is when you combine have with other verbs in Present Perfect.
You can say:
I have asked her.
Have you finished your work yet?
He has spoken.
You cannot say:
I got asked her.
Got you finished your work yet?
He got spoken. (theoretically possible, but its sense would be completely different - if there is a sense at all.)
In some other cases, have and got can replace each other - but not always.
I have a problem.
I got a problem.
I have got a problem.
You will have a problem.
You will get a problem.
I have a brother.
I got a brother.
I have got a brother.
Anyway, I believe this topic is way more complex, so I will be glad to see other opinions