Frequentist is a special case of Bayesian. The Frequentists have much more mature tools, because they've been working on them since Gauss's day. Bayesians (especially less experienced ones) may claim that that Frequentists are old school, outdated, and don't teach undergrads the new Bayesian way of doing things.
Bayesian methods are more flexible and general, but are often slow (computationally), and can be too flexible. A Bayesian can prove anything. Frequentists have trouble eliminating some biases (because their tools aren't as flexible), but also have trouble purposely (or subconsciously) biasing their results.
I'm not going into the specifics of the methods here, just the source of their disagreements.
First of all, Bayes predates Gauss. It is inaccurate to suggest frequentist statistics predate Bayesian statistics.
Second, neither is a special case of the other.
Third, neither Bayesian methods nor frequentist methods are inherently "more flexible."
Bayesian statistics set up a model and infer model parameters by applying Bayes' rule to the data. Bayes' rule is an indisputable rule of probability.
For any model parameter b, the result of applying a bayesian model is a probability distribution for b given the available data. Criticisms of bayesian models typically center around the fact that you must use a "prior distribution" indicating the modeler's beliefs about b before seeing the data. Bayesian statisticians have a number of responses to this criticism (that some people find compelling, and others do not).
Frequentist methods build models that are justified by their properties in repeated resampling. For instance, a frequentist method is "unbiased" if, given multiple hypothetical samples, it would on average produce the correct parameter b. Frequentist hypothesis testing reports the probability of observing specified data given some assumption about b.
A standard criticism of frequentist methods is that a modeler wants a probability distribution for an unknown parameter given the known data... rather than knowing the probability of observing the realized data given some assumption about the parameter.
I would describe this breakdown as misleading at best.
Frequentists are not a special case of anything. Standard frequentist arguments make absolutely no sense from a Bayesian perspective. For instance there is no prior in which such a thing as repeated significance testing errors can possibly exist. Conversely frequentists can legitimately point to a lot of things that Bayesians do which are at best highly questionable. Such as picking a default prior that gives massively high probability to clearly unlikely scenarios. (Yes the Bayesian replies, but we could pick better priors. But, the frequentist retorts, in practice you don't.)
I think that it is best to learn both, then do whatever makes most sense for your circumstances.
Frequentist is a special case of Bayesian. The Frequentists have much more mature tools, because they've been working on them since Gauss's day. Bayesians (especially less experienced ones) may claim that that Frequentists are old school, outdated, and don't teach undergrads the new Bayesian way of doing things.
Bayesian methods are more flexible and general, but are often slow (computationally), and can be too flexible. A Bayesian can prove anything. Frequentists have trouble eliminating some biases (because their tools aren't as flexible), but also have trouble purposely (or subconsciously) biasing their results.
I'm not going into the specifics of the methods here, just the source of their disagreements.