In clinical settings and situations where probabilities really matter, its a better fit.

I studied stats at Duke which is a Bayesian academy. Almost every problems come from regimes with small sample sizes. Given that Duke houses the largest academic clinical research organization globally, having a stats and biostats department with this bent is useful: samples are tiny in clinical trials compared to most big data settings.

The biggest problem with the whole Bayesian regime IMO is that as the data gets larger its selling point vanishes. If your data is big or is normal (mean-based statistics), a frequentest/bootstrapped CI approximates the Bayesian CI anyway.

Furthermore, many us work in settings where we're trying to sell toothpaste: we don't need the Bayesian guarantees that an insurer might.