$-3 \leq m \leq 4$ and $-10 \leq n \leq 9$

To maximize $|m - n|$, we maximize or minimize $m - n$:

Maximum: $m = 4$, $n = -10$: $|4 - (-10)| = 14$

$\text{Greatest possible value of } |m - n| = 14$