Frequentist statistics interprets probabilities as frequency of a certain outcome among a large number of trials. (Actually the limit as the number of trials tend to infinity).
A Bayesian interpretation of probability is "degree of belief". What is the probability that tomorrow there will be rain ? Well, a priori I believe it is about 33%. However the sky is clouded today, which has 50% probability to cause rain the next day, etc. etc. So Bayes rule kicks in and I can make a decision whether or not to carry umbrella.
For a frequentist, probability of a single event is meaningless: 33% of all days in the past have been rainy. 50% of all days following cloudy days have been rainy, etc. . . If there were a large number of parallel worlds, in x % of them there would be rain, etc. etc.
Disclaiming that I'm more of a statistics enthusiast than a real statistician, here's how I understand the schism.
Bayesian (subjective) and Frequentist (objective) are two schools of thought about how statistics should operate. One of the best ways to think about the difference is to call Bayesians pragmatic and Frequentists rigorous.
In that face of trying to quantify something you don't know, Bayesians take the stance that if you just say something — even if it's incorrect — and then keep adjusting it as more data comes in then you'll eventually have a valid statement. The idea is that even if your model is only so correct, at least you have one. Unfortunately, no one can actually prove that sort of thing actually works all of the time. It just seems to.
Frequentists faced with this situation instead try to understand the reason why something is happening and then model it from the beginning. By considering these rigorous models and testing them against data they eventually build a resilient model for the unknown which validates. That is, unless they don't, in which case Frequentists are kind of out of luck.
So when you're talking about statistics, which is all about trying to model things you don't understand, Bayesians and Frequentists get up in arms all the time because they each have something to call foolish about one another.
The coolest part is that this sort of schism is being reflected in the world of physics as well where Frequentists are in the Newtonian/Einsteinian school but, bit by bit, that worldview is being shaken up by Quantum.
Bayesians claim to be more rigorous than frequentists, living or dying by the theorems of probability theory, rather than using a toolbox of ad-hoc tools. For example. And quantumness has very little to do with it.
