It is not difficult to understand the differences between 'Doppler redshift' and 'Cosmological redshift' conceptually. But how do astronomers distinguish them when doing observations?
I think there are duplicates of this, but couldn't immediately locate them.
The answer is you cannot tell observationally whether a single redshift measurement is caused by the expansion of the universe or by something moving away rapidly.
However, if one wished to interpret the ensemble of redshifts that we see in a non-expanding universe, then you must place us at the centre of the universe. You would then have to explain why more distant galaxies are moving faster and why at very large distances, the simple Hubble law breaks down (and the microwave background and the primordial elemental abundances and so on).
Cosmological redshift can be interpreted as the accumulated Doppler redshifts between infinitely many frames of reference, each locally at rest and moving away from us at a certain velocity. So you cannot distinguish the two because they are basically different cases of the same thing.
However, as it says in @Rob Jeffries' answer above:
- when looking at many galaxies in the same region of the Universe, and they all show velocities distributed around a certain average velocity away from us, and
- when we see that this velocity-distance relation holds true across the sky,
we conclude that this average velocity is the expansion velocity of the Universe at that distance, and the deviations are Doppler shifts due to local proper motion of the individual galaxies.
Although I'm not an expert, I believe the simple answer is:
1) you can not tell them apart in individual cases
2) very simply, statistically you can pick up (apparently - quite clearly and obviously) the big picture. "Standard candles" can help and, very simply, overall averages set a picture.
I believe this is a dupe question .. Differentiating the gravitational redshift and the cosmological redshift?