I think of differential topology as the study of the properties of smooth manifolds that do not depend upon a metric. One might say that it is the study of invariants of smooth manifolds under diffeomorphsms. Differential geometry on the other hand studies the properties of smooth manifolds that do depend upon a metric, particuarly a smooth metric tensor on a smooth bundle over the manifold.
These two definition are overly restricitve but capture the basic idea of the difference.
Classical subjects in Differential topology are
- What are the cobordism classes of smooth manifolds?
- what are the topological invariants of smooth manifolds?
- when does a combinatorial manifolds admit a smooth structure?
- What are the smooth structures on a manifold?
Differential geometry covers a vast array of subjects