I'm not used to read this website, so I post my remark very late, after having randomly googled this page. I think that one of the reason why diffeological spaces are better than general sheaves is the possibility to consider infinitesimal Fermat extensions:
Giordano P. " Fermat reals: nilpotent infinitesimals and infinite dimensional spaces" . Book in preparation, see http://arxiv.org/abs/0907.1872, July 2009.
Giordano P. "The ring of fermat reals", Advances in Mathematics 225 (2010), pp. 2050-2075.
Giordano P. "Infinitesimals without logic", Russian Journal of Mathematical Physics, 17(2), pp.159-191, 2010
Giordano P., Kunzinger M. "Topological and algebraic structures on the ring of Fermat reals". Submitted to Israel Journal of Mathematics on April 2011. See http://arxiv.org/abs/1104.1492
Giordano P. "Fermat-Reyes method in the ring of Fermat reals". To appear in Advances in Mathematics, 2011.
Giordano P. "Infinite dimensional spaces and cartesian closedness". To appear in Journal of Mathematical Physics, Analysis, Geometry, 2011.
This is a new theory, and I post this answer also because I think it is not known. The basis of the theory is a surprisingly simple extension of the real field containing nilpotent infinitesimals. We start from the class of little-oh polynomials, i.e. functions $x:\mathbb{R}_{\ge 0}\rightarrow \mathbb{R}$ that can be written as $x(t)=r+\sum_{i=1}^{k}\alpha_{i}\cdot t^{a_{i}}+o(t)$ as $t\to 0^+$, where all the coefficients and powers are reals. Then, we introduce the equivalence relation between little-oh polynomials $x\sim y$ iff $x(t)=y(t)+o(t) \text{ as }t \to 0^{+}$. The ring of Fermat reals $ {}^\bullet\mathbb{R}$ is the corresponding quotient set. The theory of Fermat reals has been developed trying always to obtain a good dialectic between formal mathematics and intuitive interpretation. Even if there are several theories of infinitesimals, only a couple of them always have this intuitive interpretation, and this contradicts the idea that (rigorous) infinitesimals are a strong support to guess some mathematical truths. Of course, Fermat reals take strong inspiration from smooth infinitesimal analysis, even if, at the end, it is a radically different theory. In fact, in the corresponding ring of scalars, which extends the classical reals, we have nilpotent infinitesimals of every order, infinitesimal Taylor's formulas (analogous of the Kock-Lawvere axiom), powers, roots of (nilpotent!) infinitesimals, logarithms, a total order relation, and the ring is also geometrically representable, so that we can finally state that infinitesimals are no longer ghosts of departed quantities.
It is also very interesting to note that its mathematical definition uses only elementary analysis and Landau's little-oh notation, without requiring a background in mathematical logic. In particular, the model is so simple that can be studied directly in classical logic without any need to switch to intuitionistic logic. On the other hand, this extension of the real field is generalizable both to finite and infinite dimensional manifolds (more generally to diffeological spaces). The extension ${}^\bullet(-): \mathcal{C}^\infty \rightarrow {}^\bullet\mathcal{C}^\infty$ (here $\mathcal{C}^\infty$ is the category of diffeological spaces and $ {}^\bullet\mathcal{C}^\infty$ is the category of Fermat spaces, which are defined similarly to diffeological spaces) is functorial and has very good preservation properties: a full transfer theorem for intuitionistically valid sentences is indeed provable (the "true" logic of smooth spaces is always intuitionistic!). Several applications to differential geometry has been already developed: e.g. tangent vectors to any diffeological space $X \in \mathcal{C}^\infty$ can be defined, similarly to SDG, as smooth functions of the form $t:D\rightarrow {}^\bullet X$, where $ D :=\{h\in {}^\bullet\mathbb{R}|h^2=0\}$ is the ideal of first order infinitesimals and where ${}^\bullet X\in {}^\bullet\mathcal{C}^\infty$ is the Fermat space obtained extending $X$ with new infinitesimally closed points. At present, we are developing several notions of differential geometry in this framework and are trying to extend the theory so as to include infinities and generalized functions (distributions).