What role does differentiability play in Topology?

I might as well answer this.

Differentiability has nothing to do with general topology. The definition you are using:

Topology is the study of "deforming" figures according to certain rules

is a useful motivation for topology, but it is just a starting idea. A truer definition for topology is "studying continuity."

Differentiability makes sense only if we can talk about rates of change, and general topology doesn't even have a notion of "direction."

In general topology, we can define the concept of a manifold. An $n$-manifold is a space that looks "locally" like Euclidean $n$-space. For example, a sphere is a $2$-manifold, because if you are at a point on the sphere, the other points near can be given coordinates. Given pairs of real numbers, you can map them (continuously) to a unique points near you, and that map reaches all points near you. We call these local coordinates a "chart."

This still doesn't get us differentiation on the sphere. If you and I are on two points near each other, you and I also might have wildly different charts. In particular, for the points that are "near" both of us, what looks like a "smooth" curve or function to you in your coordinates might look "not smooth" in my coordinates. For "smooth" to make sense on the whole manifold, these charts have to agree about what is smooth.

So we have to add a structure to a manifold to make it differentiable. Essentially, if $S$ is the set of points near both you and me, then the map from my coordinates for the points in $S$ to your coordinates for the points in $S$ has to be differentiable for us to be able to talk about "smoothness" on the manifold.

I would say this is an extremely deep question, and I admit I cannot give a good/ short answer. But to begin with one might want to look at Topology from the Differentiable Viewpoint, which does not assume much from the reader, and continue with Differential Forms in Algebraic Topology, which might need a little background in algebra and topology.

According to my limited knowledge, nothing answers your question better than Milnor and Bott & Tu.

Topological topology (A joke I've heard from Milnor!) is intimately, and unavoidably related to Differential topology. There are many topological invariants that can be obtained (or even are defined) in differentiable category. And, being topological invariants, they will definitely tell A LOT about the topology of the manifold.

Example: The concept of the degree of a map helped Brouwer prove his famous fixed-point theorem, way before techniques or even proper definitions of homology and algebraic topology were set straight. Milor's amazing book "Topology from differentiable Viepoint" has a proof of this theorem, and it is interesting that once the theorem is proved for a differentiable map, it immediately implies the continuous case by approximation argument.

Another example is: The Euler characteristic of a smooth manifold is zero iff there exists a nowhere zero vector field on it!

Of course, the examples above is only a funny and trivial manifestation of the fact that not only differentiable topology is a great realm of study on its own, it is a necessary and significant must for a topologist as well. (Even the Poincare conjecture was proved through differentiable setting, on which analysis is doable.)