hypothesis testing involves induction, with prior theories that are tested against each other. It's all or nothing. One theory or the other. Nothing in between. It's similar to a criminal trial, where there are two legal theories, either the defendant is guilty or not guilty. the initial theory, the null hypothesis is a presumption of innocence, tested against the prosecutor's theory of guilt. It's one or the other, with no half guilt or partial guilt. It's similar to pregnancy. Either the person is pregnant or not. There is no half or partial pregnancy.
This is contrasted with Bayes Theorem, a deductive approach, where new evidence is used to modify the original theory. The modification is a middle approach, between the old evidence and the new evidence. You don't reject a null hypothesis or fail to reject it, You just modify it. Bayes Theorem is not a an all or nothing approach like statistical hypothesis testing. In a legal context, the defendant is sort of found partially guilty, not as innocent as the null hypothesis, the presumption of innocence, but not as guilty as the prosecutor's alternative hypothesis. A jury is stuck with 2 choices acquit or find the defendant guilty. Often they feel the defendant is not as guilty as the prosecutor claims, but they don't believe a not guilty is appropriate either. So they compromise and find the defendant guilty of some lesser included charge. IF it's a murder case, the jury might come back with manslaughter or negligent homicide, less serious charges, if they're allowed that choice. Or the jury sometimes can't agree at all, and fails to reach a verdict. That's a "hung" jury or a mistrial, comparable to statistical hypothesis testing where the evidence is very borderline and you don't know whether to reject or fail to reject the null hypothesis.
Philosophers of science prefer the Bayesian approach over hypothesis testing, as it avoids the "induction Problem"
Hypothesis testing requires some alternative' theory to test that is different from the current theory, usually the alternative theory is based on some science, finance, legal or other prior known evidence. The Bayesian approach doesn't require any alternative theory. The Bayesian apriori probability is like the null hypothesis that gets modified by new evidence, into an aposterior probability, but without any alternative theory.
Hypothesis testing seems less efficient if the alternative hypothesis has no substantial theoretical justification, or when it's more likely there are a spectrum of possibilities rather than just two possible alternatives. IF the problem is whether A or B is true, and there are no other alternatives, hypotheis testing looks more efficient than Bayesian statistics. A legal example is when the defendant pleads not guilty due to false identification or alibi defenses. Either he is fully guilty or someone else did it. Just one of the two posssibilities, nothing in between. Either total innocence or total guilt. In that scenario, the jury wouldn't tend to compromise on lesser charges.
Hypothesis Testing has the "Induction Problem" along with other problems such as confounding variables, spurious correlations, confusion of correlation with causation, unreliable outliers, and arbiturary choice of the margin of error. Never choose the margin of error after you see how it will after the choice of rejecting the null hypothesis. That's cheating.
But Bayesian statistics has the "Old Evidence Problem," where modifying the original hypothesis becomes just ad hoc changes to a theory that maybe invalid. Example is the heliocentric vs. geocentric theory. Each anomalous fact or evidence was used to salvage the geocentric theory with more epicycles, making it look equally valid as the geocentric theory. Major difference was simplicity, Ocaam's Razor, where the more simple theory is preferred. Otherwise the continuously modified geocentric theory, that the sun revolves around the earth, had equally good predictions of the positions of the planets and sun.
Another resolution is Karl Popper's "crucial experiment." Take the predictions of two hypotheses, compare them, to see which works better, but use a prediction that is most unexpected. Not just comparisons of the overall data. Isaac Newton's astronomical theories were used to predict the return of Halley's comet, far into the future. When it happened long after Newton & Halley had died, if virtually was proof positive of Newton's theory. It needed no ad hoc modifications to fit new evidence.
