What are the main differences between these three quantum theories?
Classical mechanics and quantum mechanics are subfields of the branch of physics called mechanics, that deal with two realms of size, the big and the small, respectively. The border between big and small has not be scientifically defined yet, but almost every object we deal with can be assigned to a respective group (i.e. galaxies, stars, planets, people, ants, and dust particles are all big. Atoms, quarks, photons and electrons are all small). Classical mechanics is a set of physical laws and their corresponding equations that describe/govern the motion and interaction of big bodies within the universe. These equations are Galilean invariant which means they do not apply to non-inertial reference frames. Classical mechanics is sometimes still called Newtonian mechanics because it's basis is on the work of Isaac Newton. Classical mechanics is an approximation of General Relativity in a weak gravitational field. Quantum Mechanics is a set of physical laws and their corresponding equations that describe/govern the motion and interaction of small bodies within the universe. Quantum mechanics as we know it is the Copenhagen Interpretation which has a set of several main principles . There are two widely taught formulations of QM, the wave formulation (Schrodinger), and the matrix formulation (Heisenberg).
Physics frameworks lead into each other as follows:
Mechanics leads to Statistical Mechanics
Statistical Mechanics leads to Thermodynamics
Then there is the quantum frameworks, which in their limiting case become the corresponding classical ones
Quantum Mechanics, limiting case Mechanics
Quantum Statistical Mechanics, limiting case Statistical Mechanics
Then there is Quantum Field theory, which developed as a way to calculate crossections and is appropriate for particle physics, i.e. small dimensions, high energies.
Mechanics leads to Neuton's Gravity Theory
General Relativity has a limiting case Neuton's Gravity Theory
There is no consistent quantum gravity outside of String Theory, which is the frontier being studied now.
String Theories have as limiting cases General Relativity and Quantum Field Theory.
All capitalized theories are "physics conceptual frameworks used for calculations"