In particular, it follows that if one cannot write $f,g$ in such a way with nontrivial (that is degree at least 2) $F$ and $G$ then the original polynomial is irreducible [except if one of $f,g$ is constant, but this should not be the case in view of the question]. (I am not sure how to check this most efficiently, but if the polynomials are not too complex, just starting from maximal terms and inferring conditions on the rank and then working ones way down could be a viable, though likely not optimal, strategy.)
If one wishes to have more complete information one is now raises faced with the question when a polynomial $$F(u)-G(v)$$ is reducible(ir)reducible.
Various interesting results on this problem where obtained (see below for some recent of them), but if one wishes to have an answer for specific polynomials one migt get by via using not these results, but general irreducibilty criteria for polynomials in two variables or easier to apply criteria for this polynomial.
For the former the question pointed out in a comment by Camilo Sarmiento seems like a good resource (I reproduce the link for simplicity http://mathoverflow.net/questions/14076/irreducibility-of-polynomials-in-two-variables ).
For example the Ehrenfeucht criterion mentioned there might allow to directly exclude further cases. Also the Eisenstein (like) criteria could be quite useful. Also the paper by Davenport, Lewis, Schinzel mentioned in my comment below should contain some test, but as I have no access to the paper I do not know what exactly.
Now for more recent results specifically on this problem:
In particular, Pierrette Cassou-Noguès and Jean-Marc Coveignes (Factorisations explicites de $g(y)-h(z)$, Acta Arith. 87 (1999)) based on earlier work by Fried and Feit and others established an explict finite set of pairs of polynomials such that any pair $(F,G)$, with $F,G$ indecomposable and not linearly related (this means $F(x)$ is not of the form $AG(ax+b)+B$ with constants $A,a,B,b$ and $A,a$ non-zero, to avoid corner cases), with $F(u)-G(v)$ reducible is weakly linearly realted (I skip the def, but similar to lin related) to one of themin a simple way.