Fundamentally, the disagreement between frequentists and Bayesians concerns the definition of probability.

For frequentists, probability only has meaning in terms of a limiting case of repeated measurements. That is, if I measure the photon flux $F$ from a given star (we'll assume for now that the star's flux does not vary with time), then measure it again, then again, and so on, each time I will get a slightly different answer due to the statistical error of my measuring device. In the limit of a large number of measurements, the frequency of any given value indicates the probability of measuring that value. For frequentists probabilities are fundamentally related to frequencies of events. This means, for example, that in a strict frequentist view, it is meaningless to talk about the probability of the true flux of the star: the true flux is (by definition) a single fixed value, and to talk about a frequency distribution for a fixed value is nonsense.

For Bayesians, the concept of probability is extended to cover degrees of certainty about statements. Say a Bayesian claims to measure the flux $F$ of a star with some probability $P(F)$: that probability can certainly be estimated from frequencies in the limit of a large number of repeated experiments, but this is not fundamental. The probability is a statement of my knowledge of what the measurement reasult will be. For Bayesians, probabilities are fundamentally related to our own knowledge about an event. This means, for example, that in a Bayesian view, we can meaningfully talk about the probability that the true flux of a star lies in a given range. That probability codifies our knowledge of the value based on prior information and/or available data.

The surprising thing is that this arguably subtle difference in philosophy leads, in practice, to vastly different approaches to the statistical analysis of data. Below I will give a few practical examples of the differences in approach, along with associated Python code to demonstrate the practical aspects of the resulting methods.