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.