Perhaps the best way to think about the difference is as one of emphasis.
"Proof-based Calculus" would have the goal of doing Calculus but would justify the methods of Calculus (integration and differentiation) by proving their validity.
"Analysis" would have the goal of developing the theory of Calculus (and more than just Calculus)* at least partly for its own sake.
Consider the example of the real numbers. "Proof-based Calculus" is concerned with the real numbers only because it will need them for doing Calculus. "Analysis", on the other hand, treats the set of real numbers as an object worthy of study without having to consider further applications.
*Analysis is also much broader in scope than Calculus.
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.
Best Answer
They both play off of each other nicely but, in general, differential topology is the study of the topology of a manifold given some smooth structure on that manifold. Calculus on a manifold however, is the study of the manifold and the smooth structure itself.
Perhaps the discrepency is best told by how these subjects differ in how they calculate (co)homology on the objects of their study (although this difference may appear subtle). In differential topology, the first algebraic tool you'll encounter is Morse homology which uses the differential structure (specifically the location and type of critical points) to define a sequence of groups associated to the manifold. These groups turn out to not actually depend on the differential structure on the manifold but only the topology, and so Morse homology is a topological invariant on manifolds which admit a smooth structure. This is exhibited normally by showing an isomorphism between the Morse homology and cellular homology of smooth manifolds.
In the realm of calculus on manifolds however, the first algebraic invariant you'll meet is De Rham cohomology. This sequence of groups is much more abstractly defined as a collection of equivalence classes of differential forms on the manifold and in my opinion depends more heavily on the differential structure from the beginning. Ultimately, De Rham cohomology is also a topological invariant and does not depend on the smooth structure either, so the discrepancy between how reliant on the differential structure the construction is is in their formulation. Something can also be said for their use. De Rham cohomology sheds light on much information about how integration of forms on the manifold acts. Morse homology on the other hand tells us how a smooth gradient-like vector field on our manifold will act and how critical points of various indices interact.
At the end of the day, sometimes groups of theorems don't fit in to nice boxes as we'd like them to and there can be considerable overlap between two 'subjects' (which, let's face it, is a label that humans have artificially constructed for a collection of results and 'objects'). This is definitely one case, and often it's best to study both in parallel (perhaps starting with calculus on manifolds and Riemannian geometry as they have a slightly lower entry-level).