Identify the values for m and n and the binary operation.

The given binary operation is $m \Delta n = \frac{1}{m^3} - \frac{1}{n^3}$.
We need to evaluate $(\sqrt{3} - \sqrt{2}) \Delta (\sqrt{3} + \sqrt{2})$.
Let $m = \sqrt{3} - \sqrt{2}$ and $n = \sqrt{3} + \sqrt{2}$.

Calculate the product $mn$:

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

Since $mn = 1$, we can write $n = \frac{1}{m}$.

Substitute $n = \frac{1}{m}$ into the definition of the operation:

Given $m = \sqrt{3} - \sqrt{2}$.
First,