$L_1$ is regular, having regular expression $(aa)^*(bb)^* +a(aa)^*b(bb)^*$ , either both $m$ and $n$ are even or both are odd then $m+n$ will be even
in case of $L_2$
$L_2=\{a^mb^n\: | m-n=4\}$
$=\{a^mb^n\: | m=n+4\}$
$=\{a^{n+4}b^n\}$
$=\{aaaaa^nb^n\}$
$L_2$ is CFL.