We show that the Potts model on a graph can be approximated by a sequence of independent and identically distributed spins in terms of Wasserstein distance at high temperatures. We prove a similar result for the Curie--Weiss--Potts model on the complete graph, conditioned on being close enough to any of its equilibrium macrostates, in the low-temperature regime. Our proof technique is based on Stein's method for comparing the stationary distributions of two Glauber dynamics with similar updates, one of which is rapid mixing and contracting on a subset of the state space. Along the way, we prove a new upper bound on the mixing time of the Glauber dynamics for the conditional measure of the Curie--Weiss--Potts model near an equilibrium macrostate.