If they are different, what are the typical problems in each that do not fall on the other category? Or are the mutually exclusive or does one completely capture the other?
A function over finite strings is called computable if it can be computed by a program (more formally, a Turing machine).
A set of finite strings is computable if it membership problem in that set (given $x$, is $x \in L$?) can be decided algorithmicaly (more formally using a Turing machine).
They are used for different kind of objects.
The word "computable" can be used for a set. When we say that a set is computable we mean that the set is decidable (which is equivalent to saying that the characteristic function of the set is computable).
It makes no sense to say a function is decidable.