Without loss of generality, we may assume that $x$ is the largest of $x, y, z$; then we have:
$x^{m+n} + y^{m+n} + z^{m+n} - x^m y^n - y^m z^n - z^m x^n = (x^m - z^m)(x^n - y^n) + (y^m - z^m)(y^n - z^n) \ge 0$.
The minimum is $0$ and is obtained if $x = \max\{y, z\}$ and $y = z$, so if $x = y = z$.

The maximum is achieved for $x = 1$, and its value is
$$1 + y^{m+n} + z^{m+n} - y^n - y^m z^n - z^m = 1 - y^n(1 - y^m) - z^m(1 - z^n) - y^m z^n \le 1,$$
with equality only if one of $y$ or $z$ is $0$ and the other one is $0$ or $1$.

Therefore, there are $6$ equality cases, namely
$$(x, y, z) \in \{(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1)\}.$$