Is there a difference between vibration and waves?
This is possibly an exercise in semantics. Or maybe a real curve-ball :)
Waves can be seen abstractly as energy in motion.
With the water wave , the vibration is the up-down motion as the wave moves towards, say, the beach. The amount of energy associated with that wave is totally physical, and based in classical physics. The wave (ie its particles) moves at a velocity that is determined by the prevailing conditions. We can imagine adding the vibrational and translational motion of the wave particles, and their potential energy, in order to arrive at some conclusion as the the energy or the power of the water wave.
A photon is also a wave, with a frequency and wavelength: but it is also a particle. It's energy is given by
I'm only answering because I would really appreciate a really clever answer to this question :)