What’s the relationship between gain (dB) and power (watts)?

Wouldn't an increase in the gain setting be the equivalent of turning up the volume knob, as a default setting? (using more power by default?)

I have an amp with independently adjustable gain for each of 4 inputs. I don't use it, but my understanding is that it will simply adjust (offset) the default volume (power) level for a given input. This makes sense if you wanted to level out some inputs that might be varying in volume, but why the global gain switch? Gives you a better/preferred starting point for volume.. due to input or speaker behavior?

@rols

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder.

I thought that increased damping factor was part of the reason that higher-power amps have more speaker control (grip), but I'm probably missing other factors; interesting topic.