Q.For sound waves, the Doppler formula for frequency shift differs slightly between the two situations:

In vacuum there is no medium to provide an absolute frame of reference for light, so only the relative velocity between source and observer matters — that symmetry is what makes the two Doppler cases identical; a physical medium breaks this symmetry. …

Step 1: Why sound's two cases differ

For sound waves, the propagation medium (air) provides a physically real, preferred reference frame — the medium is 'at rest' in some frame, and the wave speed is fixed relative to the medium. Whether the source moves through this medium or the observer moves through it are physically distinct situations (they are only equivalent when the speeds involved are small compared to the wave speed), which is why the exact classical Doppler formulas differ slightly for the two cases.

Step 2: Why light in vacuum is different

Light does not require a medium to propagate — the vacuum does not define any preferred frame of reference (this is a foundational postulate of the special theory of relativity: the speed of light in vacuum is the same, $c$, in every inertial frame). Because there is no medium to be 'at rest' relative to, there is no physical distinction between saying 'the source moves and the observer is at rest' versus 'the observer moves and the source is at rest' — these two descriptions refer to exactly the same physical situation, differing only by choice of reference frame, and only the relative velocity between source and observer is physically meaningful.

Step 3: Consequence for the Doppler formula …