Let $x = 2001^{1002} - 2001^{-1002}$ and $y = 2001^{1002}  + 2001^{-1002}.$  Find $x^2 - y^2.$
We have that
\begin{align*}
x^2 - y^2 &= (x + y)(x - y) \\
&= 2 \cdot 2001^{1002} \cdot (-2 \cdot 2001^{-1002}) \\
&= \boxed{-4}.
\end{align*}