- divergences are defined specifically on probability distributions
- distance metrics can be defined on other types of objects too
All distance metrics between probability distributions are also divergences, but the converse is not true—a divergence may or may not be a distance metric
Unlike metrics, divergences are not required to be symmetric, and asymmetry is important in applications.
Secondly, divergences generalize squared distance, not linear distance, and thus do not satisfy the triangle inequality, but some divergences (such as the Bregman divergence) do satisfy generalizations of the Pythagorean theorem.