jmc

The graphs intersect when the $y$-values at a particular $x$ are equal. We can find this by solving $$3-x^2+x^3=1+x^2+x^3.$$This simplifies to $$2(x^2-1)=0.$$This has two solutions, at $x=1$ and $x=-1$. The $y$-coordinates for these points are $$1+1^2+1^3=3$$and $$1+(-1{)}^2+(-1{)}^3=1.$$The difference between these values is $\boxed{2}$.