Note: I am aware of philosophical differences between Bayesian and frequentist statistics.
For example "what is the probability that the coin on the table is heads" doesn't make sense in frequentist statistics, since it has either already landed heads or tails -- there is nothing probabilistic about it. So the question has no answer in frequentist terms.
But such a difference is specifically not the kind of difference I'm asking about.
Rather, I would like to know how their predictions for well-formed questions actually differ in the real world, excluding any theoretical/philosophical differences such as the example I mentioned above.
So in other words:
What's an example of a question, answerable in both frequentist and Bayesian statistics, whose answer is different between the two?
(e.g. Perhaps one of them answers "1/2" to a particular question, and the other answers "2/3".)
Are there any such differences?
If so, what are some examples?
If not, then when does it actually ever make a difference whether I use Bayesian or frequentist statistics when solving a particular problem?
Why would I avoid one in favor of the other?