1- Is one of these statistics (KDE's p-value, KS statistic or the two-tailed p-value) recommended for my needs? If so, why?
Your needs as expressed do not seem to be sufficiently clearly defined as to differentiate between them. They both test for a difference in distribution.
2- What is the difference between the "KS statistic" and a "two-tailed p-value"?
The two sample Kolmogorov-Smirnov statistic is the largest difference in ECDFs for the two samples:
(The data here is the same data I generated for your other questionfor your other question. Here the A sample is red and the B sample is blue.)
The height difference in ECDFs at x=35 is 1/6 or about 0.1667 (indeed anywhere in $[34.50717,35.32252)$ ), the same as the value produced by calculating the statistic:
> ks.test(A,B)
Two-sample Kolmogorov-Smirnov test
data: A and B
D = 0.1667, p-value = 0.6228
alternative hypothesis: two-sided
The meaning of the p-value is as for any hypothesis test - the probability of obtaining a statistic at least as unusual (in this case, at least as large) if the null hypothesis were true.
3- Will the difference in the number of elements in each set affect the outcome of these statistics?
No, the KS test, and (to my understanding) the KDE-based one both handle different sample sizes.
Here's approximate data values, in case anyone needs them
> print(A,d=3)
[1] 41.34 25.92 55.30 50.06 75.67 3.03 61.81 34.51 34.33 9.62 94.95 24.73
[13] 30.41 11.77 25.13 90.75 12.62 36.14 56.91 29.76 15.34 62.58 33.03 36.44
[25] 47.90 66.01 42.49 18.21 31.58 58.30 17.63 70.81 73.86 46.63 10.24 12.02
[37] 47.14 15.56 80.27 12.76 33.61 52.08 41.64 13.19 32.96 64.21 81.15 32.37
[49] 33.79 40.43
> print(B,d=3)
[1] 39.43 57.93 72.91 12.81 3.76 39.02 56.02 40.28 30.25 75.31 2.46 81.44
[13] 11.74 9.32 60.85 75.39 44.58 62.05 53.33 63.63 29.90 31.41 59.82 50.37
[25] 41.17 49.49 20.34 35.32 33.82 35.47
>
All that said, I recommend you consider whuber's words most carefully. There's a lot of good advice packed into very few words.
