jmc

We have $g(4)=3\cdot 4+5=17$, so $f(g(4))=f(17)=17+3=20$. We also have $f(4)=4+3=7$, so $g(f(4))=g(7)=3\cdot 7+5=26$. Therefore, $f(g(4))-g(f(4))=20-26=\boxed{-6}$.