The Kolmogorov-Smirnov test is designed for two independent samples. You need a test for paired observations. I do not think anyone has ever proposed a KS test analog for paired observations. Also, it is a very cautious test, that is, it has low power in many circumstances.
I suggest a paired nonparametric test, either a sign test or a Wilcoxon signed-rank test, both based on the differences between the two scores for each subject. The latter is more common and has strong power for observations that have a normal distribution and a wide range of non-normal distributions.
If the two measures are strongly correlated then it is possible that the sign test would be quite powerful, possibly more so than the Wilcoxon signed rank test. Both tests are based on permutations of the observations: the signed differences for the Wilcoxon signed rank test and only the signs of the differences for the sign test.
Friedman's test is usually used for three or more measurements. If there are only two measurements, it gives the same p-value results as the Wilcoxon test.