I don't know much about statistical uses of Bayesianism but can say something opinionated about the underlying philosophy.
From a philosophical point of view, Bayesianism is fairly weak and lacks argumentative support. The underlying idea of probabilism - that degrees of belief have to be represented by probability measures - is in my opinion wrong for many reasons. Basically the only well-developed arguments for this view are Dutch book arguments, which make a number of questionable assumptions. Besides, priors are also often not known. As far as I can see, subjective utilities can only be considered rational as long as they match objective probabilities, i.e., if the agent responds in epistemically truth-conducive ways (using successful learning methods) to evidence and does not have strongly misleading and skewed priors.
I also reject the use of simple probability representations in decision theory, first because they do not adequately represent uncertainty, second because they make too strong rationality assumptions in the multiattribute case, and third because there are good reasons why evaluations of outcomes and states of affairs ought to be based on lexicographic value comparisons, not just on a simple expected utility principle. Generally speaking, Bayesians in this area tend to choose too simple epistemic representations and too simple value representations. The worst kind of Bayesians in philosophy are those who present Bayesian updating as if it was the only right way to respond to evidence. This is wrong on many levels, most notably by misunderstanding how theory discovery can and should work.
In contrast, frequentism is way more cautious and does not make weird normative-psychological claims about how our beliefs ought to be structured. It represents an overall more skeptical approach, especially when hypothesis testing is combined with causal models. A propensity analysis of probability may also sometimes make sense, but this depends on analytical models and these are not always available.
There are good uses of Bayesian statistics that do not hinge on subjective probabilities and any of the above philosophical views about them, and for which the priors are well motivated. But the philosophical underpinnings are weak, and whenever I read an application of Bayesian statistics I first wonder whether the authors haven't just used this method to do some trickery that might be problematic at a closer look.
I'd be happy if everyone would just use classical hypothesis testing in a pre-registered study with a p value below 1%.