Define the given values for m and n and the binary operation.

Let $m = \sqrt{3} - \sqrt{2}$ and $n = \sqrt{3} + \sqrt{2}$.
The binary operation is defined as $m \Delta n = \frac{1}{m^3} - \frac{1}{n^3}$.

Calculate the product $mn$:

Using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:

Since $mn = 1$, it implies that $m = \frac{1}{n}$ and $n = \frac{1}{m}$.

From $m = \frac{1}{n}$, we have $m^3 = \left(\frac{1}{n}\right)^3 = \frac{1}{n^3}$. Therefore, $\frac{1}{m^3} = n^3$.
From $n = \frac{1}{m}$, we have $n^3 = \left(\frac{1}{m}\right)^3 = \frac{1}{m^3}$. Therefore, $\frac{1}{n^3} = m^3$.
Substitute these into the definition of the binary operation:

Using the binomial expansion $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$:

Using the binomial expansion $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$:

The value of $(\sqrt{3} - \sqrt{2}) \Delta (\sqrt{3} + \sqrt{2})$ is $\boxed{22\sqrt{2}}$.