I have two data sets to compare, each containing 20 azimuths (no vector length). I'm struggling to find an appropriate test to use in R to compare and look for significant similarity. I have been using the "CircStats" and "Circular" packages in "R", but cannot figure out how to properly use the Watson's Two Sample test for homogeneity, and don't even know if it's the right test to use. Any suggestions?
A Watson Two test is a kind of rank sum test that accounts for true circularity, in that it doesn't assume any normal, or otherwise parametrized distribution. Since you have azimuths they won't span a full circular range, but probably only a 90 or 180 degree range. However, since you didn't tell us this, full circularity might still be present and warrant a Watson Two test.
In case you don't have full circularity, a "simpler" or at least more "standard" test might be equally appropriate as a Watson Two test or more so. If there is no full circularity, I would first see if the data is normally distributed (e.g. with a Kolmogorov-Smirnov test) and go from there. If it is normally distributed, there is little to argue against a t-test and otherwise, a non-parametric test, such as a Wilcoxon signed rank test would probably also be fine.