Of course the two fields have a lot in common but I wouldn't say they deal with the same kind of problems. Sure there are problems you can approach with both algebraic and differential topology, for instance you can prove Brower's fixed point theorem (https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem) with both. The algebraic topology proof uses singular homology while a differential topology proof can use something like approximation by a smooth function and Sard's theorem (for sure you can find these proofs in any book on the subjects). Broadly speaking differential topology will care about differentiable structures (and such) and algebraic topology will deal with more general spaces (CW complexes, for instance).
They also have some tools in common, for instance (co)homology. But you'll probably be thinking of it in different ways. An algebraic topologist is more interested in singular homology (https://en.wikipedia.org/wiki/Singular_homology) and a differential topologist in deRham cohomology (https://en.wikipedia.org/wiki/De_Rham_cohomology) or Morse homology (https://en.wikipedia.org/wiki/Morse_homology).
But let me give two examples of problems that are clearly from one area and not the other. The first is the existence of homeomorphic but not diffeomorphic differential manifolds. This is clearly a topic of differential geometry. The first such example was found by Milnor who showed the existence of exotic structures on $S^7$ i.e. a manifold homeomorphic but not diffeomorphic to $S^7$ (https://en.wikipedia.org/wiki/Exotic_sphere).
On the other hand an algebraic topology problem (or more specifically a homotopy theory one) is the computation of the homotopy groups of spheres (https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres). Sure spheres are also smooth manifolds but now we only care about their topology. And this sort of computations use very algebraic techniques like spectral sequences (https://en.wikipedia.org/wiki/Spectral_sequence) and the "weird" Eilenberg-Maclane spaces (https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space).
You can also take a look at arxiv and compare the tags "Algebraic topology" and "Geometric topology" (roughly the same as differential topology, I guess) to get an idea of how different the modern stuff is in both fields.
To wrap it up, I think that the best way to understand these fields and appreciate them is to learn at least the basic ideas in both. About books, I first learned algebraic topology by Hatcher and differential topology by Hirsch and I really liked both of them. Milnor's book "Topology from a differentiable point of view" is also very nice.