This is different. In math, it doesn't matter which method you're using: all correct methods that yield an answer will yield the same answer. So, the best method is merely the easiest to apply, or the one that'll yield the answer fastest.
Bayesian methods are similar: given the entirety of the information you have at your disposal (prior + data), there is one and only one posterior probability distribution over all possible conclusions. No matter how you approach the problem, applying probability theory correctly will yield the same answer.
The only drawback is that doing it correctly is often computationally intractable. So we take shortcuts, such as Monte Carlo, hence approximating the correct method, possibly yielding different results depending on the exact nature of the approximation.
Frequentist methods however can yield different conclusions depending on who you approach the problem, before you even approximate anything. In part because it sometimes takes into account irrelevant information, such as the researcher's state of mind at the time of the experiment (did I stop at 100 trials because I just got statistical significance, or did I stop at 100 trials because that was decided in advance?).
To me, such insanity is hard to fathom. Are frequentist methods that stupid? The glimpse I have got so far indicate they might be.