jmc

Substituting the definition of $f$ and $g$ into $h(x)=f(g(x))$, we get $h(x)=ag(x)+b=a(-3x+5)+b=-3ax+(5a+b)$.

Since $h^{-1}(x)$ is given by adding 7 to $x$, the inverse of $h^{-1}$ is given by subtracting 7. Therefore $h(x)=x-7$. We can test this by substiting $$h(h^{-1}(x))=(x+7)-7=x.$$Combining these two expressions for $h$ we get $$-3ax+(5a+b)=x-7.$$From here we could solve for $a$ and $b$ and find $a-b$, but we notice that the substitution $x=2$ gives $$-6a+(5a+b)=2-7$$or $$b-a=-5.$$Therefore $a-b=\boxed{5}$.