In a hallway discussion at the University, somebody claimed that atomic and molecular Chemistry is just Physics because the laws that govern the former are those well-studied by the latter. I have often held similar views, but when somebody else poses the same same, it is easier to be assess the idea critically.
There is a whole Wikipedia article on this distinction between the two. The discussion is what is commonly accepted, that the two fields have differences in scope and approach, with blurry boundaries at various points. I think this is a valid answer from a non-technical perspective, which will satisfy almost anyone that there are questions that physicists are not interested in, and other questions that chemists are not interested in.
However, I want to give a technical answer to the distinction, one that I think generalizes through the entire hierarchy of sciences, starting from Physics, going through Chemistry and Biology, moving on to Psychology, and then branching into Economics and Sociology, etc. My observation is not why these fields diverged in the first place, rather, here is a simple rule that can be used easily post-hoc find the division between fields. The rule is not super strong, and I don't think it helps to darken blurry boundaries.
The insight comes from computational complexity, applied to the mathematical models associated with the theories that are used in these fields. For example, the fundamental equations of physics are those of classical and quantum physics, from which most physical phenomena that physicists are usually interested in answering, can be inferred. However, these equations can be analytically solved for the set of relatively simple systems, and can be numerically solved for a set of slightly more complex systems. In the case of quantum mechanics, you can basically calculate the behavior of simple atoms or of crystals. Anything more complex or a system which is too large, and the computational complexity of the equations blows up and becomes out of the reach of current computers.
At this point, to be able to model the system, you have to make some serious approximations and write some simpler equations. These are the equations of applied Physics, or with further approximations, the equations of Chemistry. And there is your divide. Physics is at the top the staircase. Computational complexity has forced you down one step into Chemistry. At this point, it becomes difficult to answer the questions Physics asks, because the approximations you made threw away information from the equations. Your new equations can now answer a different set of questions, and therefore, both your scope and approach have changed.
In certain systems, the system size can gradually be increased, or you can make a series of ever more serious approximations. In such systems, the step between Physics and Chemistry will look more like a slope, or a blurry boundary.
The above applies to any other pair of adjacent fields. At some point, the equations of Chemistry become too complicated and you have to step down to the even simpler equations of Biology, and then you make a big jump to behavioral psychology, and then to Economics. While, humans have different reasons for making those jumps, the background reason is that computational complexity of the underlying phenomena makes it easy for humans to make that choice.
One final point. While I have characterized the difference as computational complexity, you can just as well make similar arguments with experimental complexity - the resources required to do an experiment. Note that Physics labs usually have the most complex equipment, while near the bottom of the staircase, Economics experiments require basically no specialized equipment.