jmc

Applying the difference of squares factorization individually to the first pair of terms and the last pair of terms, we have $$\begin{array}{rl}1003^2& -997^2-1001^2+999^2\\ & =(1003+997)(1003-997)-(1001+999)(1001-999)\\ & =2000(6)-2000(2)\\ & =\boxed{8000}.\end{array}$$