Definition of smooth manifolds in Milnor's characteristic classes

Question

I just started reading Milnor's characteristic classes, but I got a bit annoyed with his definition of manifolds, smooth maps, tangent spaces etc. since he embeds everything into Euclidean spaces. Is there any crucial inconvenience that I will encounter in this book if I just use "usual" definition of those concepts (ones essentially formulated in the language of locally ringed spaces)? Also, I am not perfectly sure that an embedding into an Euclidean space uniquely determines a differential structure on a topological manifold, i.e., I am concerned that his definition only says existence of differential structures but does not fix one. Thank you in advance.