No, there is no difference between 'I've got problem' and 'I have a problem', between 'Have you got any sisters or brothers? and 'Do you have any sisters or brothers?, or between 'I haven't got any money' and 'I don't have any money'.
Note that we don't normally say 'I have got xx'. We usually make the contraction 'I've' and say 'I've got xx.'
In these situations, 'have' indicates possession of some kind, and the meaning is exactly the same. The 'got' form is a little more informal than the 'have' form. We mainly use the 'got' form in spoken English and in informal written English. We don't use the 'got' form in formal written English, such as academic essays or business correspondence.
Also note that we don't use 'got' when the verb have replaces another verb. For example, we can say 'I have a shower at 7 am every morning'. In this situation, you can't use 'got'. You can only say 'I have..' in this situation.
I hope that helps.