Technically speaking, there is a slight difference between the two. A slim chance indicates there is still some hope, while a fat chance basically means no way on God's green Earth will this ever happen.
I believe that the expression "slim chance" came first, and eventually when someone wanted to be emphatically sarcastic they used the opposite of "slim" and designated something impossible as a "fat" chance.
Of course, this is just one of thousands of examples of confusing expressions in English. How do you pre-heat an oven? An oven has only two states: heated or unheated! Why do we drive on parkways and park on driveways? If olive oil comes from olives, where does baby oil come from? etc.