Great question! The main difference is sarcasm. If your friend says "Will it rain today?" and you answer "fat chance!" you're being sarcastic--like, "that's never going to happen." But if you say "there's just a slim chance it will rain today," no sarcasm at all.

here it is in a video:

and again:

It's one thing that those two are the same, another thing that hard working and hardly working are the opposite

Please stop making people fool. There is no such thing of slim chance. Being a professional teacher you should give an academic reference from books.

I like the sarcasm twist to this answer

"fat chance", is probably meant more as a funny way of saying - "no way!!"

"slim chance" is more a statement of fact

nice!