As the title says I want to quantify the difference in the rates of sequence change between genes. When I measure the mutation rates between my subgenomes I can determine a significant but slight difference via wilcoxon test (the distributions are gaussian but with tails). This ends up being true for many of the different types of mutations I measure, but visually I can see that different types differ in the magnitude of their difference. What is the best way to quantify the magnitude of the difference? I don't think comparing the p-values is right, and the subtraction of the median seems crude but maybe I am overthinking this.
"the distributions are gaussian but" -- the inclusion of 'but' means "the distributions are not Gaussian".
If you think there's a location-shift*, you can produce an estimate of its size from the Wilcoxon, since it has an associated estimate (Hodges-Lehmann) and confidence interval for the population quantity. (It will only estimate the difference in medians if it is a location shift alternative. But then, equally, if means exist, it will also estimate the difference in means.)
* you can adapt the approach to estimate a scale-shift if you think that's more like what's going on.
What you should use depends on what you want to find out and what you're prepared to assume.