This is only a partial answer, and you may know it already since I mentioned it recently at the nForum, but for completeness, here it is. Theorem C2.2.13 in Sketches of an Elephant shows that the following are equivalent for a category C:
- C is the separated objects for a Lawvere-Tierney topology on a Grothendieck topos (hence reflective in that topos)
- C is the category of sheaves for one Grothendieck topology on a small category which are also separated for a second Grothendieck topology
- C is a locally small, cocomplete quasitopos with a strong-generating set
- C is locally presentable, locally cartesian closed, and every strong equivalence relation is effective
A category of this sort is called a Grothendieck quasitopos. The third characterization seems most similar to what you're looking for. I doubt you can get away without some generating-set condition, since it seems very unlikely that the (complete, cocomplete, locally small) quasitopos of pseudotopological spaces (for example) can be reflectively embedded in a topos.
What I don't know is whether one can put conditions directly on a reflective subcategory of a topos, analogous to left-exactness of the reflector, to guarantee that it is of this form. The reflector for separated objects preserves finite products and monics, but I have no idea whether that would be sufficient as a characterization.
I think there are some theorems which are easier to prove in the diffeological framework, or as you say: for which the proof reveals more conceptual reasons. For example this one ?
Proposition Let $X$ be a connected diffeological space, let $\omega$ be a closed 2-form on $X$. Let $P_\omega \subset {\bf R}$ be its group of periods. If the group of periods $P_\omega$ is (diffeologically) discrete (that is, is a strict subgroup of $\bf R$) then there exists a family of non-equivalent principal fiber bundles $\pi : Y \to X$, with structure group the torus of periods $T_\omega = {\rm R} / P_\omega$, equipped with a connexion form $\lambda$ of curvature $\omega$. This family is indexed by the extension group ${\rm Ext}({\rm Ab}(\pi_1(X)), P_\omega)$.
This theorem is a generalization of the classical construction of the prequantization bundle of an integral symplectic (or pre-symplectic) manifold, that is the ones for which $P_\omega = a {\bf Z}$. Why such a generalization is interesting? Well, here are some comments:
1) The only condition for the existence of such "integration structures" is that the group of periods is diffeologically discrete, which is hidden in the classical construction by some technical hypothesis (countable at infinity or analog statements).
2) The space $Y$ is a quotient of the space ${\rm Paths}(X)$ on which the form $\omega$ is lifted modulo the action of a "Chain-Homotopy" operator (actually what is built by quotient is a groupoid and the bundle $Y$ is just the "half-groupoid"). So, the diffeological space ${\rm Paths}(X)$ is a master piece of this construction (but almost everywhere in diffeology), and the fact that diffeological spaces support differential forms (in particular ${\rm Paths}(X)$) with the whole tools of Cartan calculus is fundamental.
3) The generality of this theorem involve essentially "irrational tori", since in general the quotient $T_\omega$ is of course not a Lie group.
The last point illustrates why irrational tori are important in diffeology: or you accept these objects or you give up this (kind of) theorems. You may note that such a theorem doesn't exists in the restricted category of Frölicher spaces since irrational tori are trivial there. You may be happy with just the integral case, but in my opinion you miss a lot by not taking the whole generality of the construction, and putting fences where they do not exist.
I may give some other examples where diffeology give a shortcut for known classical theorems, and by the way extend them to objects which do not belong to the category of manifolds.
Here is an example of a more conventional theorem: the homotopic invariance of De Rham cohomology. Differential forms and De Rham cohomology are well defined concept in diffeology, they apply in particular on space of paths of diffeological spaces, spaces of smooth maps, quotients etc.
We use here the Chain-Homotopy operator
$$
K : \Omega^p(X) \to \Omega^{p-1}({\rm Paths}(X)) \quad \mbox{which satisfies} \quad K \circ d + d \circ K = \hat 1^* - \hat 0^*,
$$
where $\hat 1$ and $\hat 0$ are the maps defined from ${\rm Paths}(X)$ to $X$ by $\hat 1(\gamma) = \gamma(1)$ and $\hat 0(\gamma) = \gamma(0)$.
Proposition Let $X$ and $X'$ be two diffeological spaces, let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$, let $\alpha$ be a closed $p$-form on $X'$. The pullbacks $f_0^*(\alpha)$ and $f_1^*(\alpha)$ are cohomologous.
Proof Let $\varphi : X \to {\rm Paths}(X')$ be the map defined by $\varphi(x) = [t \mapsto f_t(x)]$. The pullback by $\varphi$ of the identity $K(d\alpha) + d(K\alpha) = {\hat 1^*}(\alpha) - {\hat 0^*}(\alpha)$ gives $d(\varphi^*(K\alpha)) = f_1^*(\alpha) - f_0^*(\alpha)$. $\square$
This is an example of simplification/generalization of a classical theorem by short-cuting the proof through diffeology. Here also the space of paths of a diffeological space, and the Chain-Homotopy operator, are crucial. May be something more fundamental is hidden behind that. Enxin Wu a Dan Christensen student is working on a possible Quillen model based on diffeology, it will give maybe some lighting on this question?
BTW Frölicher spaces is equivalent to the full subcategory of what we call "reflexive diffeological spaces" (a work in progress with Y. Karshon and al), the ones whose diffeology is completely defined by the real smooth maps. They are the "intersection" of the category {Diffeology} and the category {Sikorski}. There are nice examples and counter-examples which illustrate the difference between these categories.
Best Answer
I want to give two simple observations about diffeological spaces that might provide a partial answer to your question.
1) We have the following inclusions of full subcategories $$Mfd \subset Diff \subset Sh \subset PSh$$ where $Mfd$ is the category of smooth finite dim manifolds, $Diff$ are diffeological spaces (i.e. concrete sheaves on cartesian spaces), Sh are sheaves on cartesian spaces and $PSh$ are presheaves on cartesian spaces. The last two inclusions are reflexive.
Lets us first have a closer look at the inclusion $Sh \subset PSh$. Following the same vein of argument as above, there is a priori no reason to work with $Sh$ instead of $PSh$ since both categories are equally nice (topoi) and the definition of a presheaf is clearly simpler than that of a sheaf. But there are some colimits in $Mfd$ that we really like, namely the coequalizer diagram correspoding to an open cover $(U_\alpha)$ of a manifold $M$. Under the inclusion of $Mfd$ into $PSh$ this is not a coequalizer anymore, in other words: If we glue open sets in $PSh$ together we do not get the same thing that we get when glueing together as manifolds. This defect is exactly cured by the sheaf property. That means restricting to the smaller subcategory $Sh \subset PSh$ the colimits change such that gluing of open sets behaves as nice as in manifolds. The punchline is that the restriction to $Sh$ provides the category with the "right" coequalizers of open sets.
Now lets turn towards the inclusion $Diff \subset Sh$. The situation is exactly the same as before. Limits in $Diff$ are computed as Limits in $Sh$ (and hence also $PSh$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.
Hence one could argue that diffeological spaces have the right "geometric" colimits and sheaves do not. The price is of course that we exclude some interesing "spaces" like the sheaf of diffential forms and loose the property that the category is a topos.
2) If we want to "make" geometry over diffeological spaces it turns out that there are two possible definition of principal bundles:
a bundle over a diffeological space $M$ is a morphism to the stack of bundles over finite dimensional manifolds. This means that we have a family of bundles over each plot together with coherent isomorphisms. Note that this type of bundle is determined by its pullback to finite dimensional spaces. This is equivalent to have a diffeological space $P \to M$ together with a free transitive on fibers action such that the quotient map $P \to M$ is a surjective subduction (i.e. becomes a submersion on each plot). To get those type of bundles we have to equip diffeological spaces with the Grothendieck Topology of subductions.
a bundle over a diffeological space $M$ is a space $P \to M$ with a free, transitive on fibers, action such that it is locally trivial, where locally refers to the underlying topological space of $M$. This is the type of bundle which people consider in the world of $\infty$-dimensional manifolds. To get this we have to take the grothendieck topology of morphisms that are surjective and admits local (in the topology) sections. Hence therefore we really need the underlying topological space.
I do not prefer one of the two possible Grothendieck Topologies, but the second one is closer to what people have done in the $\infty$-dimensional setting. And one can show that the universal bundle $EG \to BG$ for a compact Lie-group is of this type (of course one has to find diffeological models of $BG$ and $EG$).
The first topology has an obvious analogue on the category $Sh$ of all sheaves but the second crucially uses the underlying topological space of a diffeological space.