Find the positive difference between the two solutions to the equation $\displaystyle\sqrt[3]{4 - \frac{x^2}{3}} = -2$.
We get rid of the cube root sign by cubing both sides.  This gives us $4-\frac{x^2}{3} = -8$.  Solving this equation gives $x^2 = 36$, so $x=6$ or $x=-6$, so the positive difference between the two solutions is $\boxed{12}$.