First of all, the concept of a "manifold" is certainly not exclusive to differential geometry. Manifolds are one of the basic objects of study in topology, and are also used extensively in complex analysis (in the form of Riemann surfaces and complex manifolds) and algebraic geometry (in the form of varieties).
Within topology, manifolds can be studied purely as topological spaces, but it is also common to consider manifolds with either a piecewise-linear or differentiable structure. The topological study of piecewise-linear manifolds is sometimes called piecewise-linear topology, and the topological study of differentiable manifolds is sometimes called differential topology.
I'm not sure I would necessarily describe these as distinct subfields of topology -- they are more like points of view towards geometric topology, and for the most part one can study the same geometric questions from each of the three main points of view. However, there are questions that only make sense from one of these points of view, e.g. the classification of exotic spheres, and there are certainly topology researchers who specialize in either piecewise-linear or differentiable methods. Differential topology can be found in position 57Rxx on the 2010 Math Subject Classification.
Differential geometry, on the other hand, is a major field of mathematics with many subfields. It is concerned primarily with additional structures that one can put on a smooth manifold, and the properties of such structures, as well as notions such as curvature, metric properties, and differential equations on manifolds. It corresponds to the heading 53-XX on the MSC 2010, and the MSC divides differential geometry into four large subfields:
Classical differential geometry, i.e. the study of the geometry of curves and surfaces in $\mathbb{R}^2$ and $\mathbb{R}^3$, and more generally submanifolds of $\mathbb{R}^n$.
Local differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a local point of view.
Global differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a global point of view.
Symplectic and contact geometry, which studies manifolds that have certain rich structures that are significantly different from a Riemannian structure.
As a general rule, anything that requires a Riemannian metric is part of differential geometry, while anything that can be done with just a differentiable structure is part of differential topology. For example, the classification of smooth manifolds up to diffeomorphism is part of differential topology, while anything that involves curvature would be part of differential geometry.
Let me convert my comment to a full answer:
Unless one is (and you are not!) planning to write a PhD thesis in General Topology, Munkres is (more than) enough.
Depending on what you are planning to study later, you might encounter an issue requiring a bit more General Topology (e.g. proper maps and proper group actions, which you will find in Bourbaki), but you learn this on "need to know" basis (just pick up a General Topology book and look it up when necessary). Instead, my suggestion is to start reading Guillemin and Pollack, and Hatcher (or Massey).
In addition, you would want to (or, rather, have to) learn more functional analysis (say, Stein and Shakarchi) and PDEs (say, Evans) which will be handy if you are planning to go into modern differential topology (which most likely will require you dealing with nonlinear PDEs, believe it or not), and, in case of algebraic topology, - basic category theory (at least be comfortable with the language), Lie theory (at least to know the basic correspondence between Lie groups and Lie algebras), see suggestions here. Yes, General Topology is fun and there are many neat old theorems that you will learn by studying it in more detail, but you have to prioratize: Life is short and your time in graduate school is even shorter.
Best Answer
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.