There are myriads of ways for two things to be unequal. In fact its fair to state that unless you are taking two sets of measurements of some fundamental physical constant or measuring artificially generated data, they are not going to be from the same exact distribution -- you cannot bathe in the same river twice is very true.
The way to address the problem you speak of is to think hard about the kind deviations from equality that would be most damaging to the application (were it to slip in undetected). Once you know what's the most damaging deviation, you can then select a test that is very powerful in detecting that specific type of deviation.
For example, you mention Kolmogorov(Smirnov) test. Those are daft blind at the tails of the distribution. So in case you need to catch deviations at the tail, then you can safely skip KS test, they are no good. On the other hand KS tests are very powerful around the median, if your application requires high resolution there, KS is the test you want.
The way to address the problem you speak of is to think hard about the kind deviations from equality that would be most damaging to the application (were it to slip in undetected). Once you know what's the most damaging deviation, you can then select a test that is very powerful in detecting that specific type of deviation.
For example, you mention Kolmogorov(Smirnov) test. Those are daft blind at the tails of the distribution. So in case you need to catch deviations at the tail, then you can safely skip KS test, they are no good. On the other hand KS tests are very powerful around the median, if your application requires high resolution there, KS is the test you want.