Interpretation for diffeomorphism Let $f: M \to N$ be a diffeomorphism between two differentiable manifolds.  Every book mentions that if $M$ and $N$ are diffeomorphic then they are basically the same from the differentiable point of view, ie, they are indistinguishable as differentiable manifolds.   Of course I agree about this but what I do not understand is how $f$ show this fact.   Let me explain this a bit more.  When we are dealing with a isomorphism $T: V \to W$ between two vector spaces $V$ and $W$ (I am out of the manifold context for a minute), this map tells that you can "rename" the objects from $V$, i.e., if $v \in V$ then rename it as $T(v)$ and the operations in $V$ rename it as $v_1 + v_2 = T(v_1) +' T(v_2)$, where $+'$ is the operation in $W$, thus, via the bijection you are seeing $V$ just as $W$.   That information is obtained directly from $T$ and its linear property from its definition as linear transformation.  In other words, $T$ transforms $V$ into $W$ preserving the linear structure. Now, back to the diffeomorphism $f: M \to N$ , how $f$ tells me that $V$ is basically the same as $W$?   How can I see $f$ as transforming $M$ into $N$ in such a way that $M$ and N$ are esentially the same as differeantible manifolds.   I hope I am clear otherwise let me know so I can explain it again in a better way.  
 A: A manifold is a topological space $M$ (with certain properties) together with an open cover $(U_\lambda)_{\lambda\in\Lambda}$ such that, for each $\lambda\in\Lambda$ there is a homeomorphism $\varphi_\lambda\colon U_\lambda\longrightarrow\mathbb R$ (which satisfy certain conditions). Now, the map $f\colon M\longrightarrow N$ transports this  open cover to $N$: for each $\lambda\in \Lambda$, consider $f(U_\lambda)$ a the map $\varphi_\lambda\circ f^{-1}\colon\varphi(U_\lambda)\longrightarrow\mathbb R$. This new atlas induces in $N$ a differentiable manifold structure, which is identical to its original one.
A: A differentiable manifold is a topological manifold with the extra structure (differentiable). I will split this two structures to make it more clear and continue with your analogy of "renaming stuff":


*

*A topological space is a structrue which is just a set together with a set of subsets called open sets. A diffeomorphism is to begin with a homeomorphism, and this means precisely that you are transforming one space into the other preserving the topological structure, you "rename" the objects as you said (which in this case are open sets). So you are still seeing the second space as the first one in this sense, if you want.

*A differential structure on your manifold consists on defining what maps from your manifold to $\mathbb{R}$ are what you will call smooth (just as before we defined what subsets we would call open). As with open sets there are some conditions that you want this structure to verify of course, but I won't get into that because I think it is not important for the point we are discussing. In this context, the map between two differentiable manifolds being a diffeomorphism means precisely that you are transforming one space into the other preserving the differential structure. In this case instead of "renaming" subsets we rename maps. So the diffeomorphism tells you that you can call a map from the manifold to $\mathbb{R}$ smooth if and only if the composition with the diffeomorphism is what you were calling smooth before.
Summarizing: a diffeomorphism allows you to do exactly the same as the vector space isomorphism, but only in a more complicated way if you want (because the structure is now more complicated). And you can see this by layers of structure as I suggested above (and there is even a third more basic layer which is the set theoretical, that tells you that you can rename points in $N$ by the corresponding points in $M$, which amounts to say that $f$ is a bijection).
