I was wondering what are the criteria to use Kolmogorov-Smirnov, Cramer-von-Mises, and Anderson-Darling when comparing 2 ECDFS. I know the mathematics of how each differ, but if I have some ECDF data, how would I know which test is appropriate to use?
To cut a long story short: Anderson-Darling test is assumed to be more powerful than Kolmogorov-Smirnov test.
Have a glance on this article comparing various tests (of normality, but the results hold for comparing two distribudions) Power Comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling Tests by Nornadiah Mohd Razali & Yap Bee Wah.
Anderson-Darling test is much more sensitive to the tails of distribution, whereas Kolmogorov-Smirnov test is more aware of the center of distribution.
To sum up, I would recommend you to use Anderson-Darling or eventually Cramer-von Misses test, to get much more powerful test.
Each of the three tests have better power against different alternatives; but on the other hand, all three exhibit varying degrees of test bias in some situations.
Broadly speaking, the Anderson-Darling test has better power against fatter tails than specified and the Kolmogorov-Smirnov has more power against deviations in the middle, with Cramer-von Mises in between the two but somewhat more akin to the Kolmogorov-Smirnov in that respect.
The kinds of alternatives many people find to be of interest tend to be picked up more often by the Anderson-Darling and the Cramer-von Mises test but your particular needs may be different.
The Anderson-Darling tends to suffer worse bias problems overall (for hypothesis tests, bias means that there are some alternatives you're even less likely to reject than the null -- which is not what you want from an omnibus goodness of fit test -- but it seems to be difficult to avoid in realistic situations).
A number of power studies have been done which include an array of goodness of fit tests; generally for the alternatives that they consider, the Anderson-Darling tends to come out best most often --- but if you're testing uniformity and trying to pick up say a beta(2,2) alternative, none of them do well, and the Anderson Darling is the worst.