I understand the general role of isomorphisms in mathematics. If two groups are isomorphic, they are indistinguishable by group-theoretic means. If two topological spaces are homeomorphic, they are indistinguishable by topological means. And so on.
However, I'm not quite sure if I understand the role of diffeomorphisms fully. In Differential Geometry of Curves & Surfaces, do Carmo writes that "from the point of view of differentiability, two diffeomorphic surfaces are indistinguishable." However, it is possible that two surfaces – say, the unit sphere and a sphere of radius 2 – are diffeomorphic although there are important differences: Because the diffeomorphism between them is a not an isometry, their inner geometry is different. If a curve is moved by the diffeomorphism from one of the spheres to the other, it changes its length.
So, the inner geometry is not necessarily preserved. But what is preserved? What does a diffeomorphism do that a homeomorphism doesn't do? (FWIW, I'm more interested in the intuition than in a technical description.)
Best Answer
Diffeomorphisms are the isomorphisms in the category of smooth manifolds, while isometries are the isomorphisms in the category of Riemannian manifolds.
That is to say, diffeomorphisms are under no obligation to preserve the extra structure (metric, and all the geometry that comes with it) that the manifolds might have. They do preserve the differentiable structures, in the sense that if $\varphi:M\to N$ is a diffeomorphism, then $g:N\to \Bbb R$ is smooth if and only if $g\circ \varphi :M\to \Bbb R$ is smooth.