Physics studies many other things than just atoms, electrons, and molecules. (For example, it also studies light, gravitational and electromagnetic waves, black holes, expansion of the Universe, heat, other elementary particles such as W/Z gauge bosons or Higgs bosons, and lots of other things.) Or to say the least, it studies other aspects of matter than just their composition from atoms or molecules (when it studies heat, it's about the energy carried by atoms but it is interested in completely different properties than just the arrangement of atoms inside molecules). One should realize that all the known forms of solid, liquid, or gassy matter is composed of electrons, protons, and neutrons – or from atoms and/or molecules (different levels of description).
Chemistry studies very specific things: the phenomena in which atoms in molecules reorganize into different molecules (reactions) and the structure of these molecules as an organized collection of atoms. It studies it because it wants to convert materials, liquids, or solutions to others, usually because these materials or compounds have some applications. It is a modern scientific version of alchemy that began with this research in not-quite-scientific framework. While chemistry is very advanced today, it doesn't use mathematics too much.
Physics is about the research of all aspects of Nature that are sufficiently "simple" or "fundamental" so that we may describe their behavior by accurate enough mathematics and the laws that govern this observable behavior may be translated to very accurate mathematical equations, too. It's a different motivation, different strategy, and different methodology. When systems become too complex or "composite", such as some particular complicated molecules, we usually don't expect physics to study these things that fail to be fundamental (although they may have very important applications in industry or chemistry).
In the case of atoms, physics still studies the objects using the traditional physical concepts such as position, velocity, acceleration, forces, energies – applied to electrons, atoms, and molecules. It actually wants to translate all these things to real numbers. We find out that to do so, classical mechanics must be replaced by quantum mechanics and deal with things like "wave functions" instead of positions but it still deals with real or complex functions of real or complex variables which include time.
Chemistry is much more practically oriented discipline that mostly studies the qualitative differences between compounds and how they change in the reactions.
Of course, there is a boundary between physics and chemistry. It is probably not easy to divide the subdisciplines that exist near the boundary but they include physical chemistry, quantum chemistry, chemical physics, and so on. These terms are actually different from each other and the practitioners may talk at length when they're working on physical chemistry and chemical physics but at the end, it's usually the same researchers, anyway. When it's physics, it's about the mathematical laws of Nature which matter. When it's chemistry, it's about the compounds and reactions.
Just to give you a simple analogy, physics is like "constitution" of a nation. Chemistry is a set of "laws", governing one particular types of affairs of the nation. Any law of chemistry must be "constitutionally valid". (Please don't bring amendments here, though... :) )
Physics is the "foundation". In rough terms at the very fundamental level, it mostly deals with energy, mass, and electric charge. As mentioned by "Lubos", it studies both things at atomic scale, and things at galactic scale. Its ultimate aim, (though not everyone would agree with that) is to find an equation that would explain everything. They've a name for it. Theory of Everything.
Most of the times in chemistry(exception- so called nuclear chemistry), nucleus is just there with +ve charge and mass. It doesn't involve in any reactions. Chemistry is all about electrons play.
Chemistry is a highly developed specific part of physics that studies ordinary matter, particularly molecules (as in gases), solutions (liquids) and materials (solids) and their transformations. Fundamentally it is all based on the physics of electrons and the electromagnetic field.
It originally focused on derived properties (taste, colour) that were used to identify different stuff, and as such it took much more time to quantify than physics. (A few centuries in fact.) Chemistry is still widely empirical.
Its main purpose was and still is to create new matter, and as such, it can be regarded as a kind of molecular (or "stuff") engineering. Most chemist create new molecules or novel ways to synthesize them. That's where the money comes.
Indeed the main "practical" difference between physics and chemistry in "contact" areas like Solid state, is that the chemists usually do the stuff (for instance, they synthesize new solids) while the physicists analyse more deeply their properties (or propose new emerging properties).
For historical reasons there are still some topics that are traditionally studied by chemists or engineers (particularly complex situations like porous materials and catalysis, or combustions processes, or macromolecules like biological molecules, or complex liquids, etc.) and there are some methodologies that are more widely employed by chemists (ab initio quantum chemistry, etc.) rather than physicists, but these things are likely to change.
Physics and chemistry are overlapping branches of physical science and both study matter and energy. Chemistry studies matter from the particle level (e.g., muon chemistry, proton-transfer reactions,...) up to cosmological level (cosmochemistry or chemical cosmology).
A fundamental difference is that physics is more focused to finding the universal laws of general processes, whereas chemistry focuses more on details and specific phenomena, such as what is the boiling point of this substance and why. By this reason chemistry is considered the central science because it connects physics with other natural sciences such as geology and biology.
Another fundamental difference between chemistry and physics is that the former not only describes the observable universe, but also changes it. As stated by Marcelin Berthelot, "La chimie crée son objet [chemistry creates its object]"
A third fundamental difference is that chemistry is less mathematical than physics because deals with much more complex problems. Mathematicians have traditionally avoided mathematical chemistry because of its complexity.
A fourth difference is that applied chemistry is very close to engineering. Chemical engineering or technical chemistry are examples.
You can find disciplines such as general relativity that belong to physics and disciplines such as synthetic chemistry that belong to chemistry. However, there are disciplines that belong to both. The overlap between physics, chemistry, and biology is represented in the next figure: figure http://juanrga.com/en/images/linksphyschembio.png