Solution:
Let $m \geq n$. Then $4^{m} - 4^{n} = 255$.

Let $k = m - n \geq 0$, so $4^{m} = 4^{n} \cdot 4^{k}$, and the equation becomes:
$$4^{n}(4^{k} - 1) = 255$$
Factor $255 = 3 \times 5 \times 17$.

Since $4^{n}$ is a power of $4$, i.e., $1, 4, 16, 64, \ldots$, it must divide $255$.

Let us check possible values for $4^{n}$:

- If $n = 0$, $4^{n} = 1$:
$$4^{k} - 1 = 255 \implies 4^{k} = 256 \implies k = 4$$
So $n = 0$, $k = 4$, $m = n + k = 4$.
Thus, $(m, n) = (4, 0)$.

- If $n = 1$, $4^{n} = 4$:
$$4(4^{k} - 1) = 255 \implies 4^{k} - 1 = 63.75$$
Not integer.

- If $n = 2$, $4^{n} = 16$:
$$16(4^{k} - 1) = 255 \implies 4^{k} - 1 = 15.9375$$
Not integer.

- If $n = 3$, $4^{n} = 64$:
$$64(4^{k} - 1) = 255 \implies 4^{k} - 1 = 3.984375$$
Not integer.

- If $n < 0$, $4^{n}$ is a positive rational number greater than $1$ (for $n < 0$), but $4^{n}(4^{k} - 1)$ would not be integer unless $4^{k} = 1$, i.e., $k = 0$.

If $k = 0$, $m = n$, so $4^{m} - 4^{n} = 0$, not $255$.

If $n > 0$, $4^{n}$ is $4, 16, 64, \ldots$, none of which divide $255$.

If $n < 0$, $4^{n}$ is $1/4, 1/16, 1/64, \ldots$

Try $n = -1$, $4^{n} = 1/4$:
$$\frac{1}{4}(4^{k} - 1) = 255 \implies 4^{k} - 1 = 1020 \implies 4^{k} = 1021$$
$4^{k}$ is a power of $4$, but $1021$ is not a power of $4$.

Try $n = -2$, $4^{n} = 1/16$:
$$\frac{1}{16}(4^{k} - 1) = 255 \implies 4^{k} - 1 = 4080 \implies 4^{k} = 4081$$
Again, $4081$ is not a power of $4$.

Thus, the only solution is $(m, n) = (4, 0)$.

If $m < n$, then $4^{m} - 4^{n} = - (4^{n} - 4^{m}) = 255$, so $4^{n} - 4^{m} = -255$.

So, $4^{n} - 4^{m} = -255 \implies 4^{n} - 4^{m} = -255 \implies 4^{n} = 4^{m} - 255$

But $4^{n} < 4^{m}$, so $n < m$. But this is the same as the previous case with $m$ and $n$ swapped and negative $255$.

Alternatively, $4^{m} - 4^{n} = 255$ can be negative only if $m < n$, but $4^{m} - 4^{n}$ is negative, so $255$ cannot be negative.

Therefore, the only integer solution is $(m, n) = (4, 0)$.

Thus, the answer is:
All ordered pairs $(m, n) = (4, 0)$.