Intuitively, what is the difference between homeomorphism and diffeomorphism? Significance?

Question

As the title suggests, intuitively, what is the difference between homeomorphism and diffeomorphism? Many thanks in advance. What is the significance of such a difference?

Answer 1

Homeomorphisms are the isomorphisms in the category of topological spaces and continuous functions. Diffeomorphisms are the isomorphisms in the category of smooth manifolds and functions that are not just continuous but also preserve the differential structure.

So, the difference is two-fold. First the objects are different, secondly (since every smooth manifold is in particular a topological spaces) the morphisms are required to preserve more structure.

Intuitively, a homeomorphism between smooth manifolds certainly need not preserve the differential structure. In fact, most homeomorphisms will complete disregard the differential structure. Just thing of mapping $S_1$ to $S_1$. This can be done homeomorphically in many ways, but demanding the tangent structure at each point to be mapped linearly bijectively as well required much more.