In General Topology if we a topological space $(X, \mathcal{T})$, that is homeomorphic to another space $(Y, \mathcal{K})$, there are a number of topological properties, such as compactness, connectedness, path-connectedness, that are invariant under homeomorphism.
In Algebraic Topology if we have homeomorphic topological spaces $(X, \mathcal{T})$ and $(Y, \mathcal{K})$, then we can conclude that they have isomorphic fundamental groups $\pi_1(X) \cong \pi_1(Y)$
But in Differential Topology, the question of what topological properties are preserved by diffeomorphisms seems to be something that I can't quite answer at the moment.
Certainly diffeomorphisms are stronger versions of homeomorphisms, so all the things we expect to be invariant under homeomorphisms (compactness, connectedness etc.) are also invariant under diffeomorphism.
However I would like to know if there are further topological (or perhaps non-topological) properties that are invariant under diffeomorphism?
Best Answer
Your question has two parts:
The straightforward answer is this: Diffeomorphisms preserve exactly the same topological properties as homeomorphisms do; nothing more, nothing less.
The reason for this is essentially definitional: A topological property is, by definition, a property that is preserved by homeomorphisms. Since every diffeomorphism is a homeomorphism, every topological property is preserved by diffeomorphisms. And if a particular property of smooth manifolds is preserved by diffeomorphisms but not by homeomorphisms, then it's not a topological property.
The other half of your question was
This is a more interesting question. But first you have to establish an appropriate category of spaces to work in -- diffeomorphisms are only defined between smooth manifolds, which are topological manifolds with an additional structure called a smooth structure. So the appropriate question to ask is whether there are non-topological properties of smooth manifolds that are preserved by diffeomorphisms. There are, but they're more subtle. For example, one such property for compact smooth manifolds is whether they bound (smoothly) parallelizable manifolds. This is one way that exotic spheres can be distinguished from each other.