Quite generally, whenever you ``need a moduli space'', say, polarized deformations of varieties, or spaces of morphisms, you oftentimes construct it as follows: you realize your data as some family in projective space that (over-)parametrizes your data. Then, fixing numerical invariants, you prove that some Hilbert polynomial in this projective family is constant/bounded. Then, you use the Hilbert scheme as a first approximation to your moduli problem. Usually, you've overparametrized your problem and need to take some appropriate stack/GIT quotient...

Thus, in case you're interested in arithmetic moduli, say moduli spaces of polarized Abelian varieties over $\mathbb{Z}$, you will need the fact that the Hilbert scheme can be defined over $\mathbb{Z}$.

Let me also give an application to complex geometry: when proving the existence of rational curves on (complex!) varieties, whose $K_X$ is not nef, via "bend and break", you do the following: you reduce your variety modulo positive characteristic $p$, and construct the desired rational curves on infinitely many reductions modulo $p$ using the Frobenius morphism and characteristic-$p$-methods. Then, you bound the degree of these rational curves (w.r.t. some polarization). Now, to conclude the existence of a rational curve in characteristic zero, you use the space of morphisms from $\mathbb{P}^1$ to show lifting of these curves from characteristic $p$ to characteristic zero. Here, it is essential that this space of morphisms (whose existence relies on the Hilbert scheme) is defined over some ring of integers.