[joshjob42]

Well even for simple things there's a large difference. Say you toss a coin N times and observe heads x times. What is the probability of your next toss coming up heads?

A frequentist arguably would say the question doesn't really have any meaning since probabilities are about long run frequencies of things occuring. They might do various tests or tell you the probability of that outcome under various probabilities for heads.

A Bayesian would make an initial assumption about the probability of any given probability, and then compute a posterior using the likelihood function the frequentist may have, and give you a distribution for what you should believe about the what the true probability of heads is on your next coin toss.

In general, the latter is more meaningful and informative. There's also pretty good arguments that any coherent method of representing credences is isomorphic to probability, see Cox's theorem.

[usgroup]

x ~ Binomial(N,p) and you wish to estimate p.

Here are a whole collection methods for how to estimate p and calculate a confidence interval for it: https://en.wikipedia.org/wiki/Binomial_distribution#Confiden...

One of the methods is Bayesian; the rest are not.

Not mentioned in the list, but you can also use likelihood ratio intervals calculated from a likelihood profile: another Frequentist method.

None of the methods -- including the Bayesian, requires an informative prior.