Hi Taka,
I have to tell you, I admire your question. Native speakers tend not to notice these little quirks of the language.
Slim chance means very little chance, almost no chance. It's also acceptable in formal English.
fat chance means no chance, so the meaning is not quite the same. It's for informal English only. I think that 'fat' as the opposite of 'slim' is used ironically. It's like saying, ironically, 'Oh yes, there's a good chance of that' when you know there is no chance at all.
Best wishes, Clive