Why are non-separated schemes schemes?

Question

In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X \times X$ is closed)

Nowadays, schemes are usually allowed to be separated. The question is then: Do non-separated schemes naturally occur in nature?

Answer 1

Moduli spaces and stacks tend to be non-separated. See for example this poster by David Rydh and Jack Hall. The example of the Picard scheme appears in FGA explained, Ex. 9.4.14.

Answer 2

Non-separated schemes also appear in GIT; i.e., when you want to define the quotient of a scheme by a group of automorphisms.