In this instance, they mean about the same thing. They can both be used to indicate annoyance at something that has continued for too long. Fed up can have angry connotations while tired has connotations of boredom.

Interesting to see the perspective of Americans here. Fed up can have different meanings depending on the context. In this context it has the connotation of bored, although as Richard it can also have the connotation of anger (but not really in your example). I agree with Richard that in this example the meaning of the two sentences is basically the same. Aren't you fed up of playing that game? Aren't you fed up with playing that game)? Both sound fine and natural to me. I would favour the construction with 'of' myself, but I could see myself saying both quite easily.

I would say "fed up with ..." rather than "fed up of ..." To say one is "tired of something" is a relatively neutral statement. To say one is "fed up with something" is both stronger and more colloquial.