If I have three consecutive words where each adjacent pair can be contracted, e.g. "I would have" or "You are not", is there a difference between the two possible contractions, e.g. "I would've" or "You aren't" versus "I'd have" or "You're not"? If so, when is each form preferred?
Does the preference depend on what comes after it (such as its part of speech), e.g. "You are not a fool" versus "You are not foolish" versus "You are not going to fool me?"
Whether to contract an auxiliary verb with a following negative (haven't, aren't, isn't, etc.), or with a preceding subject (I've, you're, he's, etc.) is pretty much a stylistic decision, and practice varies. With some exceptions (e.g, *amn't), either is possible.
In particular, I've not, you've not, etc. is much more common in UK English than in American English, where I/You haven't is the norm. Saying I've not seen that yet in the USA will suggest a British accent, and is the form we'd expect from an RP speaker.
As with many other constructions, saying I've not instead of I haven't, without also having an RP accent may suggest a certain formality, and possibly stuffiness, to some people.
Negation and in particular how negative contraction competes with verb contraction is a very large area of research. For example, The Longman grammar of spoken and written English devotes thirty pages to it, and there's some information in the appendix, too. Since there is so much detail there, and your question is about "I've not" vs. "I haven't", I'll answer that particular question.
Several posters suggested that "I've not" might be a Britishism. However, various researchers argued that with the verb "have" there is a very strong preference for negative contraction even in British English:
I haven't is much preferred to I've not. (The Longman student grammar of spoken and written English, 2002: 242)
They haven't finished is more common than They've not finished (Greenbaum 1992: 684 in The Oxford companion to the English language)
The Longman grammar of spoken and written English (1999) has Table A.8 in the appendix (on p. 1132). In that table, you can also see that "haven't" is "overwhelmingly the preferred choice" (p. 1131).
It's not entirely clear whether you're interested in the lexical or auxiliary "have". In any case, even in British English the lexical verb "have" sounds formal or old-fashioned (e.g. I haven't a brother). Corpus studies have shown that this use of "have" is pretty rare - the more usual being "I haven't got a brother" (BrE) or "I don't have a brother" (AmE and BrE).
In general, there is no real difference in meaning. This can be seen by expanding the contractions. There may be reasons to choose one form over the other, however. Here are two that come to mind:
Common usage: For whatever reason, some contractions with equivalent forms are just more frequent. For example, I haven't is more common than I've not. You may prefer the former to avoid standing out.
Grammatical context: You may wish to choose a contraction based on the grammatical context in which it appears. For example, contracting I have not into I haven't is correct when the have is non-auxiliary in function (e.g. "I haven't a clue"). See this answer.
"I haven't seen any" became: "I've not seen any" ... with the 1980 film "The Empire Strikes Back". Sadly, it has become acceptable literary cannon, in recent years.
