I'm aware that both it's not and it isn't are contractions of the same phrase, it is not.
Till today, I was convinced that choosing them depends on desired emphasis. This way, choosing it's not allows one to make an emphatic stress on not:
It's NOT my fault!
...and other than that it's just a matter of style, not grammar.
Recently, I was watching Monty Python's "Argument Clinic":
M: An argument isn't just contradiction.
A: It can be.
M: No it can't. An argument is a connected series of statements intended to establish a proposition.
A: No it isn't.
M: Yes it is! It's not just contradiction.
A: Look, if I argue with you, I must take up a contrary position.
M: Yes, but that's not just saying 'No it isn't.'
A: Yes it is!
M: No it isn't!
I was surprised they have never used it's not during the whole sketch.
This confused me a lot as it opposed my prior understanding: two people contradicting each other would rather use it's not.
More than that, at 01:42 they have used you did NOT and then you DIDN'T, which must indicate that the latter is more pathetic than the former.
So, how to choose a proper contraction between "it's not" versus "it isn't"? Which one is more emphatic, and how come a contracted variant looks more pathetic than the expanded one?