Define a smooth manifold to be a ringed space (a Hausdorff, second countable, $n$-manifold) that is locally isomorphic (as a ringed space) to the sheaf of smooth real valued functions on some open $U \subseteq \mathbb{R}^n$.
Similarly, a differentiable manifold is a ringed $n$-manifold locally isomorphic to the sheaf of differentiable real valued functions on some open $U \subseteq \mathbb{R}^n$. EDIT: I should also mention the sections of these sheafs are actually real valued functions defined on the open subsets of the space.
I believe I've been told that the two notions are actually equivalent; that is, a differentiable structure can be shown to be smooth. I've heard this is a difficult result.
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$\begingroup$Here's my understanding of the relationship, not being a geometer I fully expect to be either told off for misuse of language or supplanted through a better explanation of key concepts.
A differentiable manifold equipped with an equivalence class of atlases whose transition functions are all differentiable. A smooth manifold is a differentiable manifold equipped with smooth transition maps. That is to say; the deriviatives of all orders exists (i.e., it is a $C^{k}$ manifold).
For my purposes, the relations rests on this transition map 'equivalence'.
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