For example, I would hear people say in the "regime" of x>3? Does it have a different meaning/origin to the "region" of x>3? To me, "region" sounds like the more proper term.
Edit: I have heard this usage in a few context in physics, electrical engineering, and machine learning. It usually involves the professors pointing at a 2D graph and articulating about the behavior of a system/model/function in these "regimes".
The ordinary (non-scientific) meaning of "regime" has to do with governments and the laws they impose. That meaning has been carried over to scientific contexts, to refer to those domains in which certain laws or theories are valid. Thus, I might say that a certain calculation in physics is valid in the classical regime, meaning that it relies on the laws of classical physics and would not be valid in the relativistic regime (meaning when velocities are so great or gravitational fields so strong that relativity theory must be used) or the quantum regime (where the entities are so small that quantum theory must be applied). Likewise, I might refer to some range of parameters in a partial differential equation as the elliptic regime, meaning that the equation is elliptic (and I can invoke nice facts like automatic smoothness of weak solutions) when the parameters are in that range.
I hope that, if I've ever used "regime" in my own writing, I've used it in accordance with this meaning. I admit, though, that some people (possibly including me) have used "regime" just because it sounds cool.
I think it generally means a range of conditions where a system goes into a certain qualitative kind of behavior. For instance a dripping faucet drips steadily at a low rate of flow, then goes into "period doubling" where you get pairs of drips, then groups of four... and finally "chaos" where the timing seems to be random. Also solid/liquid/gas/plasma state changes at different combinations of temperature and pressure. (I'm looking for an authoritative-seeming source for this.)
The term "regime" is often used in dynamical systems where there are mutiple competing influences (or terms in the equations of motion) that would each individually cause the agent to act in qualitative different ways. A "regime" doesn't usually have a sharp boundary like $x \geq 3$, but instead refers to a vague region in parameter space where one of the terms dominates all of the other ones, so that you can either ignore the other terms in the equations of motion or treat them perturbatively (hence the analogy that the dominant term is "ruling over" the dynamics, I guess).
The different regimes are usually qualitatively different from each other, and it's typically simpler to approximately analyze the behavior deep within each regime, but there are usually fuzzy boundaries with multiple effects of comparable importance, where you're not clearly within one regime or another.
In fluid dynamics, you can have inertia-dominated regime versus diffusion-dominated regime, depending on the Reynolds Number.
Perhaps this is an example of what FlameTrap and Paul Sinclair discussed in the comments of the original question.
