This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space would mean) that would "classify" all gadgets.
The first, obvious attempt, is to set $F$ as a functor from $Schemes$ to $Sets$ like so: $F(S):=$ all the families of gadgets parametrized by $S$. Sometimes $F$ will not be representable by a scheme.
Now we may ask more cleverly, is there a stack fibered in groupoids given by $F(S):=$ all families of gadgets parametrized by $S$, with isomorphisms corresponding the various $S$-automorphisms.
It seems that people think of stacks as the "natural" object for moduli problems, but I do occasionally see papers proving the existence of a moduli stack. So let me phrase it like this:
The failure for the moduli space to be a scheme has to do with non-trivial automorphisms. What does the failure of the moduli space to be a stack indicative of? What does the existence of a moduli stack imply?