How to test if two distributions are different

I asked this question on Twitter:

what test to test if two distributions are different? I am aware of KS test. When n is large (which is common in genomic studies), the p-value is always significant. better to test against an effect size? how to do it in this context?

In genomics studies, it is very common to have large N (e.g., the number of introns, promoters in the genome, number of cells in the single-cell studies). A more concrete example is that one have two samples: …