Classical physics deals with macroscopic particles and macroscopic quantities. Like the objects we can see , feel, deform etc.. The size of the objects is more than a 100th of a millimeter or so..
Quantum physics deals with quantities and particles of the size of atoms and molecule and the sub atomic particles like proton, electron , neutron etc.
The physics and properties of particles of the size of nanometers and less is quite different from that of macroparticles that we can see with eyes. These particles have dual nature - wave like nature and particle like nature. Also these particles have high velocities like electrons when in motion. The classic physics principles of velocity, force, momentum, mass do not apply directly. The theory of relativity applies to these particles.
Quantum mechanics or physics or quantum theory are all referring to the same science subject. In quantum mechanics, the physical quantities like energy, mass, charge etc. exist in discrete quantities. They are not continuous. There is an Uncertainty principle that the mass and velocity or position cannot be determined exactly or 100% correctly. There a constant called Plank's constant denoted by h. The order of masses, energies of particles is supposed to be in that range.