How does our interpretation of a distant galaxies red shift differ if we think of it as a cosmological redshift rather than as a Doppler shift

Cosmological redshift relates to the expansion of space-time, while doppler shift relates to the relative velocity between the source and the observer.

Explanation:

Cosmological redshift relates to the expansion of space-time in the sense that the amount of the shift observed depends on the distance between the observer and the source since it's the space itself between them that is expanding and causing the redshift. Doppler shift relates to the relative velocity between the source and the observer in the sense that once light leaves the source the shift observed doesn't change, assuming the relative veolcity between the objects remains constant.