Looking at 2 strata of my data. Stratum 1 has a bigger difference
between the median and mean than stratum 2. This means that the
distribution of stratum 1 deviates even more from a normal
distribution than stratum 2 (i.e. it is more tailed). What else does
it tell us?
I think the answer is "nothing". That is, by itself, the fact that the difference between the mean and median is larger in one stratum doesn't tell you anything other than that the difference is larger. In fact, depending on what you mean by "deviates" it might not even tell you that it is farther from normal, because it could be that the strata have very different means and medians.
Suppose that the distributions had the same shape, but that one stratum had much larger values, then that stratum would have a larger difference.
As people said in the comments, the best way to examine this is to graph the data. You could start with overlaid density plots of each distribution, a parallel box plot, and a qq plot.