Assuming that $g$ is eventually positive, then
$f = O(g)$ means that $|f(x)| \le cg(x)$ for some positive constant $c$ and all sufficiently large $x$;
$f = \Omega(g)$ means that $ f(x) \ge cg(x)$ for some positive constant $c$ and all sufficiently large $x$.
Why couldn't $|f(x)|$ be also used in the definition for $\Omega$ ? Why is there this difference?