Problem 563

$2.1001,101$.
Let the larger of these two positive integers be $x$, and the smaller one be $y$, then
$$x y + x + y = 2005 \text{, }$$

which means $(x+1)(y+1)=2006$.
Notice that, the only possibilities are
$$(x+1)(y+1)=2 \times 1003 = 17 \times 118 \text{, }$$

Therefore, $x-y=1001$ or $x-y=101$.