Why is 'just' the supremum used for both 2-sample kolmogorov-smirnov test, as well as the kuiper's test, rather than a more comprehensive difference? I am looking for a method to compare two distributions of numbers (whether they differ in underlying distribution).
However, after reading thhrough respective wikipedia pages ( kuipers , 2-sample KS ) I am unsure why both tests make use of the supremum in distance on the cumulative probability axis rather than a difference metric which used the full distributions of both samples, like for example the full area of difference between the two distributions. It seems to me that this would be a more informative metric, as less of the distributions' information is discarded.
Is this an existing concept I am unaware of, or is there a specific reason why this is not desired?

I think the original reason is mathematical tractability. The distribution of the supremum is the same for any continuous null hypothesis distribution, and there's a reasonably well-behaved infinite series for it, and it was worked out a long time ago.
Other summaries do exist: the Cramér-von Mises/Anderson test uses the integral of the squared difference, and the weighted Kaplan-Meier tests of Pepe & Fleming use weighted areas between CDFs. The K-S test just got in first.
