From a data perspective whats the practical difference then? Both terms are just capturing the unmodeled error, whether that be from the linear model, or the variation not captured by a mean estimate. Is that simplification a good general understanding?
They’re only equivalent when your (assumed) model is actually true - functional form is correct, not missing anything, etc. There’s no way to be absolutely sure you set up the model correctly. And there’s no way to directly observe the \epsilon either.
Even in cases where you have a strong reason to believe in the model, things can go awry.
Eg Imagine testing Hooke’s law (F = kx + u) for the length of springs, but maybe over time, after several trials you stretch the spring too much and later trials are a little different because the “springiness” has worn out a little. The “True” model that “Nature” “uses” E(F|x) + \epsilon to “make springs a certain length” has a bunch of higher order crap in it that we’ve ignored at our own peril.
It could be “bad”. Or it could be “fine”. Depends if you are making Slinkies or rocket parts.