I'm trying to compare two distributions which are bimodal (have 2 highest maxima) and along the x-axis are skewed, one right the other one left, but given this complexity I don't know which is the most appropriate statistical test to run in order to test their significance.
In my opinion Hellinger Distance measure is best to compare the similarity between two probability distributions.
The Jensen Shannon divergence may be better for comparing theoretical distributions than Hellinger distance because the Jensen Shannon divergence is closer related to testing homogenity. If the distributions are very close these two divergences will give approximately the same results. If the distributions are discrete both Jensen Shannon divergence and Hellinger distance will fail if one has not divided the real axis into appropriate bins. For testing homogenity of two samples the Wilcoxon rank-sum test might be the best choice.
Take a look at Kolmogorov–Smirnov test which is commonly used to compare two sample distributions as long as they are continuous.
You don't test data, you test a statistical hypothesis (using some data). It is not clear which statistical hypothesis you want to test. Just as you stated: there is some additional complexity, and this means that there are several different aspects in your data that can be adressed by specific statistical hypotheses. Doing a KS-test is a kind of a "general-purpose test" for the hypothesis that the two samples are taken from the same distribution. However, to my opinion, a rejection of this hypothesis does not really provide much insight. So I think that a "good way" to make sense of your data won't be an "easy" way of simply doing some test. I think you should rather spend time and effort in finding out e.g. why your data are bimodal, and what the modes refer to.
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