From a topological point of view a homeomphism is the best notion of equality between topological spaces. I.e. homeomorphisms preserve properties such as Euler characteristic, connectedness, compactness etc. I've understood it such that diffeomorphisms are the best notion of equality between manifolds (basically a diffeomorphism is just a smooth homeomorphism), but what exactly are the properties that are preserved after a diffeomorphic map between manifolds (say M and N)? If I would guess it's a diffeomorphism from M to N preserves all the properties that a homeomorphism preserves PLUS the differentiability of M (say it's C^k). But are there any more important properties to keep in mind?