Extremely low p-values from non-parametric tests
Well, you've bumped into a well-known feature of significance tests, which is that the p-value typically goes to zero as the sample size increases without bound. If the null hypothesis is false (which can often be established a priori), then you can get as small a p-value as you wish, just by increasing the sample size. 
My advice is to think about what practical difference it makes that the distributions differ. Try to quantify that in terms of cost, either real (dollars) or abstract. Then devise a measurement for that.
I'm using Python's non-parametric tests to check whether two samples are consistent with being drawn from the same underlying parent populations: scipy.stats.ks_2samp (2-sample Kolmogorov-Smirnov), scipy.stats.anderson_ksamp (Anderson-Darling for k samples), and scipy.stats.ranksums (Mann-Whitney-Wilcoxon for 2 samples). My significance threshold to say that two samples are significantly different from each other is p = 0.01.
If these three tests return extremely low p-values (sometimes like 10^-30 or lower), then do I need to worry about something having gone wrong with the scipy functions? Are these ridiculously small p-values reliable, and can I just report p << 0.01 (p much less than my threshold)?