What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well aware of several statements made more beautiful in the language of stacks, but, I'm looking for a concrete application.

I would like to point out that stacks are "just" higher analogues of sheaves  a very basic tool to arrange structure. The same is true for topological or differentiable stacks. So I think everybody who expects amazing applications of stacks should be able to name an equally amazing application of a sheaf. (I am not saying that those don't exist!) That said, let me mention an application. In view of the fact that you didn't get any answers so far (apart from your own), I hope it's not too inappropriate to take one from my own research. It applies abelian gerbes with connection to lifting problems for principal bundles. I hope the following specifications qualify the theorem below as application: its statement does not involve any stacks or gerbes, just "basic" differential geometry. Its proof, however, is a simple composition of two gerbetheoretical theorems.
Of course some concepts that appear here would need some more explanation  but that's not the point. Let me better point out how gerbes with connection come into the picture. We employ two results from gerbe theory:
Now, define $\mathcal{L}_P$ as the transgression of $\mathcal{G}_P$. Since transgression is an equivalence of categories, it is a bijections on Homsets, and this bijection is exactly the statement of the theorem. Ok, in order to complete my claim that this is an application, I should probably mention an example where the theorem is useful. That's the case for $spin$ and $spin^c$ structures on manifolds, and I have learned about it from Stephan Stolz and Peter Teichner. In the case of $spin$ structures, $\mathcal{L}_P$ is a $\mathbb{Z}_2$bundle over $LM$ and plays the role of the orientation bundle of $LM$. Since $\mathbb{Z}_2$ is discrete, all the connections disappear and forms are identically zero. So, the theorem says that isomorphism classes of $spin$ structures on $M$ are in bijection to "fusionpreserving orientations" of $LM$. In the $spin^c$ case, a similar statement follows that additionally includes the scalar curvature of the $spin^c$structures. 


Whilst asking this, I nearly forgot that one application does come to mind: http://www.math.fsu.edu/~aluffi/archive/paper325.pdf In this paper Behrang Noohi shows how to use topological stacks to calculate the fundamental group of the quotient of a topological space by a group(oid) action by using fixedpoint data. 


I should update with a mention of some of my own results in http://arxiv.org/abs/1504.02394:
Let $G$ be a Lie group acting almost freely on a manifold $M$. Then the homotopy type of the Borel construction $M\times_G EG$ is the same as the the classifying space of a certain discrete category, whose objects are smooth tranversals, i.e. maps $f:\mathbb{R}^n \to M,$ with $n=\dim M  \dim G$ which are transverse to the $G$orbits. 

