I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) schemetheoretic argument or theorem is significantly simplified by the introduction of stacks. I'm sure there are many, but since I don't deal with stacks on a regular basis I don't encounter them as frequently, and I thought maybe some of you can enlighten me.

The DeRham space is a stack $X_{DR}$ associated to a smooth variety $X$, so that modules on $X_{DR}$ are Dmodules on $X$. This is accomplished by declaring the maps from $Y$ into $X_{DR}$ are the same as maps from $Y^{red}$ (the reduced scheme) into $X$. This has the effect of identifying points with their infinitesmal neighborhoods. The DeRham space is often most useful as a conceptual tool. However, a specific application of it was by BenZvi and Nevins, who used it (and other tools) to show that certain cusped versions $\widetilde{X}$ of $X$ had equivalent categories of Dmodules. The idea being, these cusps were identifying some of the infinitesmal neighborhoods of some of the points, and so they should be intermediate between a variety and its DeRham space. 


The classic application is Deligne & Mumford's paper proving the irreducibility of the coarse moduli scheme $\overline{M}_g$ of stable genus g curves over any algebraically closed field. They proved irreducibility first for the moduli stack $\overline{\mathcal{M}}_g$ of curves, and then inferred from this result the irreducibility of the scheme. 

