Well, I’m not up to giving my whole formal methods course here, and besides there isn’t enough space and you probably wouldn’t be interested. But since you asked, I’ll try to explain without doing violence to the subject.
I began my course with a quote from Alan Turing’s last paper, which presented a mathematical model of embryonic growth called Turing Patterns:
“This model will be a simplification and an idealisation, and consequently a falsification. It is to be hoped the the features retained for discussion are those of greatest importance in the present state of knowledge.”
Or, as George Box put it, “All models are wrong, but some are useful.”
So the issue is what Turing called “the features retained for discussion.” The Bernoulli model and the K-J model strive to represent the same physical phenomenon but retain different features, falsify different things, are wrong in different ways and useful for different things.
A good summary of the differences can be found here.
Cheers,
Earl