I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds:
Analysis on manifolds, containing: definition of manifold, tangent space (as derivations and classes of curves), vector fields, vector bundles, flows, Lie derivatives, integration on manifolds (Stokes theorem), forms, Hodge decomposition theorem
Introduction to differential geometry (for me it should be called introduction to Riemannian manifolds) containing: tensor calculus introduction, Riemannian manifold definition, connections, curvatures, geodesic, normal coordinates, geodesic completeness theorem, classification through curvature, Jacobi fields, harmonic maps.
Now, for me differential geometry was/is a theory about manifolds, so anything dealing with manifolds is a branch of differential geometry. On the other hand, I am preparing for taking part in local conference called "algebraic and differential topology". I have read: I- books references, II- books references, III- wikipedia, yet I still don't really know if differential topology is subtheory of differential geometry or is it separate theory and how is it located between differential geometry and algebraic topology. For example I expect that studies on Riemannian manifolds are part of differential geometry but would problem of classification manifolds up to diffeomorphism be a part of differential topology or geometry?
Request: I would be grateful for your characterisation of differential topology and differential geometry possibly with examples of problems, theorems present at them.