The discriminant of a polynomial is a very useful quantity but I have never seen the sign of a discriminant be useful. I know that the generalization of a discriminant from number fields over $\Bbb Q$ is other number fields is an ideal defined over the base field and clearly, in this generalized case, the sign cannot matter.
However, perhaps the sign is important elsewhere?
The sign of the discriminant of a monic polynomial $f(X) \in \mathbb R[X]$ is
$$(-1)^{r}$$
where $r$ is half the number of non-real roots of $f$.
Consider for instance a cubic monic polynomial $f(X)$ with roots $a, b, c$. Since non-real roots appear in complex conjugate pairs there are two possibilities: either all of $a, b, c$ are real, or one is real and the other two are complex conjugates.
The discriminant is
$$(a-b)^2(a-c)^2(b-c)^2.$$
This is clearly positive if all of $a,b,c$ are real. In the case where $a$ and $b$ are complex conjugate and $c$ is real, then $a-c$ and $b-c$ are complex conjugate so $(a-c)^2(b-c)^2$ is a positive real number, and on the other hand $(a-b)^2$ is the square of a purely imaginary number and is therefore negative, so the discriminant is negative.
(By the way, this is cool because you can calculate the discriminant of $f$ from its coefficients, and thus you have an easy and direct way to check whether a cubic polynomial has $3$ real roots or only $1$ real root.)