Difference between relativistic doppler effect and classical one?

When I ask for differences I am not meaning mathematical ones since that it is obvious:
We have a phenomenon p, p can be described by Galilean Relativity (GR)
and by Einsten Relativity (ER).
ER(p)!=GR(p)
The math expression that describes this phenomenon is different.
Then the doppler effect will have another mathematical expression
ER(dp)!=GR(p)
Maybe someone wants to say that GR is like a subset of ER for small velocities in reality It is not It looks similar but It is not, well let us get to the point.
Sound can be described by the classical doppler effect GR(d_e) by addition and substration of velocities though we know the velocity is constant due to the medium.
My question can we do the same with light? can we explain it mathematically with the addition of c+v, c-v?
I hope I have expressed myself clearly, I know c is a constant and We can not add velocities, what am I asking is if by using galilean relativity the doppler effect of light can be explained adding velocities, after all sound has a constant velocity and we add it.
Let us put it another way, is there any physical not mathematical proof, evidence, observation, meaning... that the doppler effect of light is different than the sound one?
Or Is It just the way It is expressed mathematically.
Because I see this applet:
http://webphysics.davidson.edu/applets/applets.html
and the phenomenon looks the same.
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