Solution:

Write $x = 100h + 10t + u$ with digits $h,$ $t,$ $u.$ Then $y = 100u + 10t + h,$ so $$z = |x - y| = 99\,|h - u|.$$

Since both $x$ and $y$ are three-digit numbers, both $h$ and $u$ run from $1$ to $9,$ so $|h - u|$ can be any of $0, 1, \ldots, 8.$ Each choice gives a different multiple of $99,$ so there are $9$ distinct values of $z.$