The Doppler effect depends on direction---it happens when the observer is moving towards or away from the source, not when the observer is moving sideways. (It happens for light---redshift and blueshift---as well as for sound.) Time dilation, on the other hand, is a genuine clock-rate-difference which is present no matter what direction people are moving in, and it's still there even if you've tried to take Doppler shifts into account.
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