Distance between regular languages
As you perhaps already know, a common metric on words is the Cantor metric, which is defined as:

$$
d(l, k) = \left\{ \begin{array}{ll}
                     0 &amp; \mbox{if } l = k \\
                     2^{-n} &amp; \mbox{where } n = \min\{i\in\mathbb{N} \;|\;l_i \not=k_i\}
                  \end{array}
          \right.
$$

Roughly speaking, if a string is a sequence of events, the distance between two strings is $2^{-n}$, where $n$ is the first time they differ. 
This can be lifted to a metric on (nonempty) languages by using the Hausdorff metric. (If you allow infinite strings, you also have to ensure that the languages are Cauchy-complete.)  

This metric shows up a lot in verification. The first reference to this that I know is Alpern and Schneider 1985, Defining Liveness. (Sorry for the absence of a link, but I couldn't find an online copy.)

Jean-Eric Pin has written a survey article, Profinite Methods in Automata Theory, in which he surveys some more general metrics, and also draws some connections to Stone duality.