I am trying to measure the dissimilarity between two empirical discrete distributions. I am aware of various distance metrics that could be used for this purpose such as Wasserstein, Bhattacharyya etc.
However, I have trouble finding out the pros and cons of using each one of the available metrics.
For example, why can't one simply use an l1 distance between the two vectors defining the distributions? Using the l1 distance does not make any sense to me especially when it comes to probability distributions but I have yet to encounter an explanation of why it is or it is not a good fit for this purpose.
The same is true for the other distances available, Wasserstein, Bhattacharyya, Hellinger distance etc. What are the drawbacks of using each one of those metrics? And why use any of them and not simply go for l1 distance.