The most important thing that you need to understand is that the verb 'have' has two different functions. Here's the rule: #1. Main verb This is where 'have' carries the meaning of the sentence, usually possession ( I have a brother) and sometimes other meanings ( I have lunch at midday). If 'have' is the main verb of a sentence in a simple tense, you form questions - "Do you have a brother?" - and negatives with the auxiliary verb 'do', the same as for any other ordinary verb. I don't have any brothers or sisters. He doesn't have a car. We didn't have a good time at the party. #2. Auxiliary verb 'Have' is the auxiliary verb in perfect constructions, followed by the past participle. In these cases, it has a purely grammatical function, so you form questions with a simple inversion, 'Have you been to London?', and you form negatives by adding 'not' or 'nt', the same as for any other auxiliary verb. I haven't been to London. He hasn't done his homework. We haven't had our dog for very long. ... Simple, isn't it? The most important exception to this very straightforward rule is the idiomatic construction 'I've got xxx'. If you say 'I've got a problem', this is an informal alternative to 'I have a problem'. The meaning is the same as #1, but the grammatical construction is the same as #2, so we form negatives by adding 'n't' : 'I haven't got a problem with that'. You do sometimes hear people use 'haven't'/ 'hasn't' followed by a direct object, especially when there is an 'any' involved, e.g. 'I haven't any money', and in certain set phrases such as 'I haven't a clue'. This is more common in British English. I wouldn't worry about these expressions if I were you. Be prepared to meet them occasionally, but don't feel that you have to copy them. When you are using the language yourself, just say 'I don't have any money' and 'I don't have a clue', as this is the more common construction worldwide. I hope that helps.

Where I'm from (Los Angeles), it would be incredibly strange to hear someone say "I haven't" when it comes to a phrase like "I haven't an idea" or "I haven't any money." It sounds super British and unnatural. On the other hand, I say "I don't have" all the time. It's the only way we say it here, although I'm sure it varies across the U.S. "I haven't" would only sound correct if I were using the verb completely differently, saying something like, "I haven't done my homework yet" or "I haven't been there before."

You have (!) to make a distinction between "have" as an auxiliary verb and as a main verb. The contraction is unusual -- but not unknown -- in the latter case, in both AmE and BrE. But a common expression in BrE -- less so in AmE -- is "I haven't got".

I haven't and I don't have - are the same, I don't know if one is more British than the other, but Americans would understand both!

Instead of 'doubt', you should say 'question' here: 'I have a question about these expressions'. Instead of 'which are the differences' you should say 'what are the differences'. As for your question, consider the following sentences for example: 1. I haven't any money with me. 2. I don't have any money with me. I'm an American, and I would say that Americans are more likely to say 2 than 1. But only just slightly. I would not be surprised to hear another American say 1.