As far as I know, the only difference between sound and vibration is that sound propagates but vibration does not. In most cases, they are the same. Please help clarify these concepts.
We will start by describing what vibration is in general and then go on to describe the human perceptible forms.
Any physical quantity which oscillates in time is a vibration or an oscillation. For example, a disturbance in a lattice of atoms gives rise to an oscillatory displacement from the mean position, which propagates along the dimensions of length. Taking such an example in gas, we talk about a disturbance in pressure. Such a change gives rise to a pressure wave through the gas. If we are looking at a disturbance or oscillation in fields, it gives rise to electromagnetic waves. These are only a few examples. A vibration can be manifested in any of the physical quantities.
Coming to the ones we can perceive, we give these vibrations different names. The perceivable pressure waves in air are sound, the electromagnetic waves that we can see are different colours in visible light. Thus, sound is a smaller part of vibrations that exist around us. Hope this clarifies the confusion.
Sound is vibration and vibrations can be sound depending on if your ear can detect them or not. The human ear can detect roughly 20Hz to 20kHz with perfect hearing. Vibrations also propagate through the material or item that vibrates so I'd say that the statement that vibrations does not propagate is incorrect. The sound however typically propagate through air, perhaps there's a misconception about maybe vibrations not propagating through the air, because at that point they'd probably be sound?