The difference is barely noticeable. Being fed up with something might mean that you are bored of it; Being sick of something is pretty much the same but with a little more emphasis. It really does depend on the context though!

fed up is probably understood to mean bored with. sick of is probably more used to mean disgusted with. they are both often used interchangeably​.

no real difference. I am sure some people will find a subtle difference. That is what happens in languages, especially English.