What is the difference between diffeomorphism and isomorphism? What is the difference between diffeomorphisms and isomorphisms? I know isomorphisms already from my abstract algebra/group theory course, and now I'm studying analysis on (sub)manifolds, where this definition is new for me. Can anybody explain what the difference between the two is? Or in other words: how can I relate those two concepts? Thanks :)

Diffeomorphism is simply a special name given to the concept of an "isomorphism of smooth manifolds".
In general, you will meet the concept of an isomorphism in many different areas. It denotes a mapping which identifies two different objects (i.e. they are basically the same object  in the category you are treating).
