> 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](http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Two-sample_Kolmogorov.E2.80.93Smirnov_test) is the largest difference in ECDFs for the two samples:

![enter image description here][1]

(The data here is the same data I generated [for your other question](https://stats.stackexchange.com/questions/77907/generate-qq-plot-for-sets-of-different-size). 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](http://en.wikipedia.org/wiki/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
    >

 

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All that said, I recommend you consider whuber's words most carefully. There's a lot of good advice packed into very few words.

  [1]: https://i.sstatic.net/EC9qW.png