Both schools use the same mathematical modeling tool, the probability measure, but they use it to model different things in the real world. Frequentists use it as a model for repeatable events like tossing a coin, while Bayesians use it to represent the degree of subjective belief to any statement.

I am more interested in the main causes of why Bayesian and frequentist statistical analysis gives different results on the same problem and the same data. Both camps, while trying to solve a given statistical problem, often make certain assumptions about the mechanism that generates the data. The core distinction and the source of fierce debate between them seems to be deciding when should we stop trusting our beliefs.

Frequentists recognize the importance of Bayes procedures as well. Wald proved that under mild conditions the set of Bayes procedures (with all possible priors) is complete. This means that for any non-Bayes procedure, you can find a Bayes procedure whose expected risk is never more, whatever the true state of nature is. Wald also proved that under the same mild conditions any minimax procedure is equivalent to some Bayes procedure with a "least favorable prior". From a Bayesian perspective, such a prior represents believing in the most unfortunate scenario, that maximizes the expected Bayes risk.

Frequentists argue that after some point (usually setting the parameters of a family of distributions) we should not trust our intuitions about the true state of nature, so we should not choose a prior distribution which reflects our beliefs. Instead, we should behave like a paranoiac who's prepared for the worst case scenario. That is why, it seems, a frequentist %95 confidence interval is almost always wider than a Bayesian %95 credence interval for example.

The resolution of this debate seems to require a scientific investigation of the following question: Under which circumstances and for which problems a person's intuitive judgements are more useful than a paranoiac minimax approach? 

If an average scientist's judgements can be shown to be superior to the minimax approach in a given domain (using frequentist techniques :) then frequentists, I believe, will be convinced to behave like a Bayesian in such domains as well. I don't know if there is any literature on this kind of a question.