Could someone explain how the relativity of simultaneity and the doppler effect are different?
They have different formulas, different units, and different uses. In fact, I can't think of a reason why there would be any confusion that they are different.
The Doppler effect is something that you actually observe and measure without regard to any theory. The relativity of simultaneity is something that you cannot directly observe nor measure unless you have a theory with which to define it.
Of course, any valid theory must predict the correct Doppler effect, as the theory of Special Relativity does. Special Relativity states that you can select different frames of reference to describe a scenario, and they will each predict the same Doppler effect, even though they can assign different coordinate times to events, which is what the relativity of simultaneity is all about.
As ghwells says, the doppler effect is "raw data", something that relates to what signals you actually get back if you listen to a broadcast and/or bounce a radar or something similar off a moving object.
The relativity of simultaneity is something you can deduce from the doppler effect with a few modest auxiillary assumptions, namely that the doppler effect depends only on the relative velocity, and that the doppler effect doesn't vary with distance, and that there is no preferred direction in space - that it's "isotropic".
Part of the deduction process involves setting up the concept of a "frame of reference" - something that you add-on top of raw sensory data to assign events locations and times to account for signal propagation delays.
If you're interested in the details, you can find them in for example Bondi's "Relativity and common sense". I gather some newer books cover this as well - Mermin was mentioned as one of the authors, but I haven't read his treatment, I assume it's very similar to Bondi's treatment.
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