 Complete metrizability is a topological property while completeness is a property of the metric.When talking about spaces with more structure than just topology, like topological groups, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology.For abelian topological groups and topological vector spaces, “compatible with the extra structure” might mean that the metric is invariant under translations.