This is a classic question that keeps recurring. And it's a good question every time it comes up, because if you really do understand the difference between gain and level, then you know something very significant about sound engineering.
Take the example of a mixing console. Each channel will have a gain control; it will also have a level control (the fader).
Superficially, both will seem to do the same thing – control how loud the signal in the channel is. But they do it in different ways.
A electronic circuit that produces gain (of which the gain control is the part you can grab hold of) increases the voltage of the signal. So the signal starts small, and through the action of the gain circuit gets bigger.
A level control (which doesn't need any electronic circuitry – just the control will do the job) reduces the level of the signal. So the signal starts off bigger than you want it, and you operate the level control to bring it down to size.
This might seem odd – the gain control comes first and boosts the signal up to higher than you need, then you reduce it to the voltage that you need using the level control (the fader).
But this has been found to work well in practice. It would be interesting to consider whether it would be feasible to build a mixing console in which the fader controlled gain. Anyone with a deeper understanding of mixing consoles than mine (I guess you would be a console designer) would be very welcome to reply.
The disadvantage of boosting the signal to higher than required, then cutting it back, is that you need 'headroom' to do this. Headroom is a range of signal levels higher than you actually need. It's like the ceiling of a room being 8 feet high (2.4 meters). Few people are that tall, but everyone enjoys the comfort of knowing that they are never going to bump their head.
Since this headroom is never used in the output, it represents a range of levels that are even further away from the noise floor than the normal operating level. So the signal-to-noise ratio of a mixing console, when operated normally, is never as good as it could potentially be. In normal use however, this is hardly noticeable.
So in summary, a gain control always makes the signal higher in voltage. A level control always makes it lower.
(And if you see a fader calibrated up to +10 dB, it means there is a 10 dB gain stage before the fader, which is itself a level control).