What is the difference between quantum mechanics and quantum physics?
There is one more term one should discuss in this thread, quantum theory.
They're largely equivalent and in most cases, each of them may be replaced by any other. S_H overstates the difference by suggesting that "quantum mechanics" is a quantum version of "mechanics" while "mechanics" is just a subset of physics. However, "quantum mechanics" isn't used just for mechanics. Quantum mechanics is used for the new set of principles that underlie all quantum theories whether or not they could be interpreted as "mechanics" or e.g. "field theory". After all, quantum mechanics shows that particle-like "mechanical" properties of elementary particles or light are always complemented by their field-like "wave" properties. In this sense, quantum mechanics does imply that "mechanics" includes the rest of physics including fundamental fields when it's done properly.
The term "quantum theory" was born around 1900 when Max Planck explained the black body radiation by the light quanta. Quantum mechanics is what was born in the mid 1920s except that later, all these advances would be considered parts of the same "revolution" that may be called either "quantum theory" or "quantum mechanics".
So one could say that all the three terms are synonyma. Physicists would almost always pick "quantum mechanics" as the preferred label for any theory that follows the postulates of quantum mechanics or for the postulates themselves. On the other hand, "quantum physics" and "quantum theory" is more likely to be used by the outsiders or in the non-expert context.
Mechanics is a subset of Physics. However, quantum mechanics or quantum physics do not necessarily mean the same thing because Physics is a very broad term where as mechanics is a specialized area of physics dealing with motion of bodies in a frame of reference.
P.S.: If you meet with any researcher in any area other than business, you would see how often they use the term physics in their research to explain a phenomenon/behavior.