- Where did this difference come from? When did it develop?

I'm not sure of the specifics but it (a) appears to be a fundamental dichotomy on ways to practice "finding a good model" given statistical mathematical foundations and (b) has been heavily politicized historically.

A lot of statistical historical practice is developing a good general purpose way of finding a good statistical model and proving that it works pretty well under some assumptions. Historically, Bayesian methods were considered taboo (perhaps because we generally lacked the ability to compute them) and so most papers were Frequentist. Very historically (Gauss) Bayesian methods were often used to generate some of the first statistical models used in physics.

- What are the basic premises that a Bayesian believes that a frequentist doesn't, and vice versa? Reason it all the way through front and back.

In basic mathematics both sides share the same beliefs, but in practice they favor different means to construct and evaluate models. See my other answer for many more details, but essentially Frequentists evaluate their models by seeing how much they diverge from reality and Bayesians evaluate them by comparing relative likelihoods of models given what they observe. This leads to wide variations in the means of constructing, elaborating, and talking about models.

- What does the B/F's model look like? What are the pieces they use, how are they arranged, what are the dependencies, how does causality flow?

A Frequentist's model can be literally anything. You might legitimately consider "the minute of the day that the mailman arrived" an estimator for "the expected time when stock A will beat out stock B three months from now" and then you use Frequentist methods to evaluate how your estimator performs. You'll also likely conclude that this estimator is terrible.

A Bayesian's model typically flows from a "generative story" which results in a massively parameterizable model which covers a huge swath of potential realities and then the Bayesian goes looking in that space for the "most probable" model.

Frequentists can use Bayesian methods if they like. Bayesians can evaluate their "most probable" models using Frequentist evaluations if they like. Good statisticians do all of the above.

- Why are the choices made by one invalid for the other's model? Where do they agree deliberately despite this?

We both want to travel from Boston to SF. I fly and get there quickly, you drive and have a great road trip. We both arrive at approximately the same place but our methods and experiences differ. For sufficiently short trips they're even identical.

More to the point, Frequentists and Bayesians disagree about their mechanisms for getting to good models. Really dogmatic Frequentists and Bayesians can disagree about "the meaning of probability" but as far as I'm concerned this has much more to do with decision theoretic policy making and education rather than mathematics.

- What are the consequences in the real world? Give me a real example on why this difference matters? "Real" meaning I don't care about dice, I care about engineering and science.

Lets say you want to model an engineering problem statistically. Frequentist methods will probably end up requiring some leaps of logic and clever tricks to get to the best result but they will also end up with at least a few algorithms you could run on constrained hardware. Bayesian methods will be easier to "plug and chug" in many parts (though they still require a lot of finesse) but the final result will almost invariably require a fast computer.

I'd compare it to integration. One school of thought is that if you're pretty clever you can integrate many things by exploiting their structure to find the antiderivative. Another school of thought is "I can answer most practical questions here through numerical integration at the end of the day, so why both finding an antiderivative?"

Both work essentially but you face very different challenges on each road and some problems can be much easier for one perspective or the other. If you're really good you have both of these tools in your toolbelt and think carefully about when to pull each one out