Given your clarification...

The two methodologies can give somewhat different results, but not as often as you might think, and the differences aren't as large as you might think. In my experience, the instances where you get radically different answers from Bayesian/Frequentist methods are quite rare, and tend toward the pathological example invented purely to demonstrate the "superiority" of one over the other.

That said, sometimes one or the other are significantly more convenient or easy to apply for a given model or type of data.