jmc

At the intersection points, the $y$-coordinates of the two graphs must be equal, so we have the equation $x^4=y=5x^2-6$, or $x^4=5x^2-6$. Putting all the terms on one side, we get $x^4-5x^2+6=0$. Factoring, we get $(x^2-3)(x^2-2)=0$, so $x^2-3=0\Rightarrow x=\pm \sqrt{3}$ or $x^2-2=0\Rightarrow x=\pm \sqrt{2}$. Thus, $m=3$ and $n=2$ and $m-n=\boxed{1}$.