Does one of these turn up the bass boost on the amp or the gain?
In a word...no.
In other words once your amp gains are set properly (which is supposed to be done with the remote bass control set full if used) the controls purpose is to actually attenuate (lower) the bass. So what would happen is if the control was set to full, and you unplugged it from the amp, the bass would not get lower..you would notice no difference.
It's main purpose is to allow the listener to have control of the bass level at thier fingertips so you can adjust the bass level to suit your tastes. Lets say you put in a song and the bass is too loud and drowning out the mids/highs. You can use the control to lower it. If you play another song and the bass is not enough, you can use the control to turn it up, but it will never boost the bass any louder then the amp would be capable of if the control wasn't used.
Since most of the decks I install have a subwoofer level control built in, I typically do not even install the control, no need for it. I find it good to use when adding a sub to a factory deck, which is getting more common these days due to many factors, or when using an entry level aftermarket deck that does not come with a sub output.
But to answer your question in a nutshell it does not change the gain settings, you need to adjust those properly with any system to avoid speaker damage, but it does lower the voltage level of the bass frequencies internally at the preamp signal to the amp as you turn it down from max on the dial.
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