OP clearly means from an observer moving at (approximately) the same velocity as the object emitting the light.

The objects aren’t really moving in that sense. The expansion of space over time is what stretches the light.

Without a fixed point of reference, what is the difference?

The difference is the expansion of space is accelerating over time. And certain wavelengths of light will be blocked by gas in the intervening space, and which light is blocked changes over time based on how red-shifted it is. Special Relativity isn't enough to explain this.