Wasserstein metric is a distance function defined between probability distributions on a given metric space M. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on M, the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved.
Jensen-Shannon divergence is a method of measuring the similarity between two probability distributions. It is based on the Kullback–Leibler divergence, with some notable (and useful) differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence is a metric often referred to as Jensen-Shannon distance.
I've seen JSD used in machine learning, but not so much Wasserstein metric even though it improves GAN. Is there a good guideline on when to use one or the other?