This is from the entrance exam for Indian Statistical Institute. $m$ and $n$ are supposed to be positive integers.
We must obviously have $m>n$, which means $m^3\geq (n+1)^3 = n^3+3n^2+3n+1$. This yields $$ 21=m^3-n^3\geq (n+1)^3-n^3= 3n^2+3n+1 $$ which means that $n$ is at most equal to $2$ (and if we allow negative numbers, $n$ also cannot be smaller than $-3$). From there you can just check every case.