What are some ways to see how different two discrete probability distributions are? I have two samples that can only take value from 1-4, in increments of 0.5, and I'm trying to compare their distributions without using Kolmogorov-Smirnov test, as that is primarily designed for continuous distributions. I am also aware of the Jensen Shannon divergence. Are there any other methods?
For general differences in distribution there's the chi-squared test. (Alternatively, you could adapt the Kolmogorov-Smirnov to the discrete case via a permutation test, which should reduce the power loss from substantial overconservatism - unless, perhaps, a lot of the distribution is focused into only a few outcomes.)
If there's a preference to detect tendency to be lower or higher you would use a test that focuses its power on detecting those sorts of alternatives. There's a number of possibilities, including a Wilcoxon-Mann-Whitney. As with the Kolmogorov-Smirnov, the heavy ties need to be dealt with. In larger samples you might consider using the normal approximation with the variance adjustment for ties, but more generally you could use a permutation test. Indeed any of the mentioned tests could be implemented as a permutation test.
Somewhat in between the chi-squared and the other options would be a Neyman-Barton-style smooth test of goodness of fit, focusing on the low-order components; a focus on only the lowest component is very similar to a z-test of means.