a diffeomorphism

If a homeomorphism is the right notion of 'the same shape' in topology — a continuous deformation with a continuous undo — then a diffeomorphism is the right notion of 'the same smooth shape'. It is a bijection between smooth manifolds that is smooth, with a smooth inverse. Two manifolds related by one are indistinguishable by any construction of differential topology: tangent spaces, vector fields, integral curves, and so on all transfer back and forth.

Precisely, f: M -> N is a diffeomorphism if f is a smooth bijection and f^{-1} is also smooth. Smoothness alone of f is not enough: the map t -> t^3 from R to R is a smooth bijection but its inverse t -> t^{1/3} is not differentiable at 0, so it is a homeomorphism but not a diffeomorphism. By the inverse function theorem, a smooth f is a local diffeomorphism near p exactly when its pushforward df_p is a linear isomorphism of tangent spaces; being a global diffeomorphism additionally requires f to be a bijection.

Diffeomorphisms are the isomorphisms of the category of smooth manifolds, so 'classify manifolds up to diffeomorphism' is the basic question of the field. The subtlety, and the reason exotic spheres are famous, is that homeomorphic manifolds need not be diffeomorphic: S^7 has 28 smooth structures, all homeomorphic to the standard sphere but pairwise non-diffeomorphic, and R^4 admits uncountably many smooth structures no two of which are diffeomorphic. So 'same topological type' and 'same smooth type' are genuinely different equivalence relations.