If you are talking about possession of something, they mean the same thing. "He has a dog." = "He has got a dog." Same with "have" used in "have to" (must) "I have to go" = "I have got to go" = "I must go." They are NOT the same if talking about an experience "She is having lunch" is correct. You cannot say "She has getting lunch." If you just use "have" instead of "have got" you will always be right.

'have got' sounds more colloquial and a tad bit rougher than 'have'. For example, if you were writing an essay for school, I would say that 'have' would be preferred to 'have got'. (And, I would say that in general, for more formal writing, it's typically better to avoid constructs involving 'get'.)

I agree with fd.