Compressor reduces a signal's dynamic range but the same can also be done by riding/automating the volume fader.
Compressors add coloration but other than that is it any different from fader riding?
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Let's compare gain/fader/wave riding (plugins or mixing engineers) and compressors
What they have in common:
What compressors do, but riding doesn't or shouldn't
What riding does but compressors don't
You are right that the basic effect that a compressor has on the signal is simply to change its gain, just as you are doing when you ride or automate the volume fader.
The 'selling point' of a device called a compressor is that for the purposes of gain reduction it does that riding for you, so that you don't have to. Additionally, a compressor can make its gain reduction much more reliably and with faster rates of change than anyone is likely to be able to do riding a fader. That's one reason that compressors may be able to add coloration - because they can respond so fast, they can change the attack characteristics of sounds they're applied to.
Essentially, the automation introduces possibilities that couldn't be realistically achieved by riding a fader, or (in a sensible amount of time) by drawing a curve.
Edit - my answer here is quite simple in what I saw as the spirit of the question - for more details please see piiperi Reinstate Monica's excellent answer.
The difference in using a compressor and riding the faders is evident in the results of each each procedure. Riding the faders results in the equal lowering of the level of an instrument or a track, that is, when the fader is pulled back, both the loud and quiet parts of the signal are attenuated equally. When using a compressor, it can be adjusted to attenuate the loudest (peak) parts of the signal to avoid overdriving the amplifier or recording device, while at the same time allowing us to boost the weaker more delicate parts of the signal into prominence. The result can be increased sustain on an instrument, with an appearance of increased overall volume, without overdriving the amplifier or recording medium. Riding the faders results in maintaining the original dynamics of the signal, whereas the use of compression reduces the original dynamics of the signal. Both techniques have their applications in music production.