This is not a question about what is special about a metric space in itself; instead, I'm wondering what sets metric spaces apart from uniform spaces?
An explanation is in order. As a parallel, when we transition from the notion of a topological space to a uniform space we get a precise notion of "x is closer to y than z". This notion allows us to formalize the concepts of Cauchy sequences (or Cauchy filters), uniform continuity, boundedness, and totally-boundedness. With each of these concepts being covered, it doesn't seem like there are any concepts that metric spaces can call 'their own.'
Inarguably, the definition of a metric space is much more concise and intuitive; however, I'm asking for a list of concepts that metric spaces can discuss that uniform spaces cannot.