jmc

We have $f(1)=5(1)+2=7$, so $f(f(1))=f(7)=5(7)+2=37$. We have $g(2)=3(2{)}^2-4(2)=3(4)-8=4$, so $g(g(2))=g(4)=3(4{)}^2-4(4)=3(16)-16=32$. Combining these, we have $f(f(1))-g(g(2))=37-32=\boxed{5}$.