Be cautious using this article to try to learn anything. Differentiable programming is not actually related to deep learning; it's another word for automatic differentiation, a technique that is very important in deep learning implementations but useful for a variety of other tasks where having gradients available for arbitrary functions is useful.

The article is correct that "Differentiable Programming" seems to be a rebranding effort that I believe just helped automatic differentiation work from the machine learning world get published in Programming Languages journals. I wouldn't read too much into it.