I recently started to read about moduli spaces. The best one can probably hope for, seems to be a fine moduli space and that is a moduli space that carries a universal family. In other words, the corresponding moduli functor is represented by a scheme. These are quite abstract notions but I think, I'm fine with the definitions.
One point that I didn't really get is what these families are really good for. For example, take the Grassmannian. It is a fine moduli space and carries the universal family $$U=\{(V,p)\in Gr(k,n)\times \mathbb P^n\mid p\in V\}$$ and we have natural projections $p: U\to Gr(k,n)$ and $q:U\to \mathbb P^n$. $p$ gives $U$ the structure of an algebraic bundle over the Grassmannian and this bundle is the tautological one (i.e. the fibre over a point is the subscheme parameterised by this point).
In 3264 and all that by Eisenbud and Harris, they compute the tangent sheaf of the Grassmannian from this which is fair enough an application of the existence of an universal family but still seems to involve other things that are special to Grassmannians.
In general, it is not clear to me, why such a family is useful. Where does it make our lifes simpler? Are there standard arguments using the existence of universal families?