We determine the generating function for the $\mathbb{S}_n$-equivariant Chow polynomials of the braid matroid $B_n$. The Chow polynomial of $B_n$ is the Poincaré polynomial of the wonderful compactification of the complement of the braid arrangement, with respect to the maximal building set. A key input to our result is the identification of this wonderful compactification with a moduli space of multiscale differentials, recently established by Devkota, Robotis, and Zahariuc. In this way, our work also contributes to the literature on topology of moduli spaces of multiscale differentials. To prove our main formula, we study the classes of these moduli spaces in the Grothendieck ring of varieties via the formalism of $\mathbb{S}_n$-spaces developed by Getzler and Pandharipande. We also give a new interpretation of the numerical Chow polynomial of $B_n$ as the Poincaré polynomial of a moduli space of genus-zero relative stable maps to $\mathbb{P}^1$.