'I have a problem' and 'I've got a problem' have exactly the same meaning.
Note that we rarely say 'I have got'. It is usually contracted to 'I've got ..'
The only difference is that 'I've got' is more informal. You shouldn't use it in academic essays or business correspondence, for example. It's fine to use it in spoken language, and in informal writing.
However, it is important to remember the following points. You can only use 'I've got' as an alternative to 'I have' ..
1. ... when the meaning is 'own' or 'possess'. For example, you can say 'I have a shower at 7 am' but you can't say 'I've got a shower at 7 am'.
2 .... in the present simple tense. Do not try to use it in any other tense.
By the way, it is not true that one is British and the other American: all native speakers of English use both forms.