Um, the folks at CERN don't agree with your assessment of frequentism in science. Their papers are explicitly frequentist, they celebrate results based on how many "sigmas" they get. P-value all the aym
However I would not expect CERN papers to make the kinds of terminological / theoretical lapses about confidence the parent thread was talking about. Papers should be rejected for that kind of error, even if you are not a bayesian.
In the case that they differ you almost always find that you have very few observations. I would argue that this 'difference' is not that exciting because it must be dominated by your assumptions, not your observations. After all once you accumulate enough observations the two methods tend to converge.
Personal conclusion: if the methods disagree work on getting more data instead of fighting over which method is better.
https://jakevdp.github.io/blog/2014/06/12/frequentism-and-ba...