The definition of a manifold in G&P is as a subset of $\mathbb{R}^n$ (as in Milnor). As I recall the the definition of diffeomorphism is such that a cube and a sphere are considered not to be diffeomorphic. This is because G&P define a map at point of a manifold to be smooth if it can be extended to a map on an open set of the ambient space that is smooth in the sense that it is a map from an open set in $\mathbb{R}^n$ to $\mathbb{R}^m$. I never understood, or saw, how this approach can be used to think about different differentiable structures on manifolds. Since there is only one differential structure on $S^2$, the definition I mention above of diffeomorphism seems to at odds with the general one, given for example in Spivak volume 1. (If anyone could explain this to me I'd be grateful. As a student I found this confusing and still do.)