A self-conscious person is acutely aware of the effect other people have on them. A self-aware person is aware of the effect they have on other people.
A self-conscious person is driven by expectations others have of them. A self-aware person is motivated by their own expectations, despite what other people expect of them.
To be self-conscious is outside-in – a servant to the external world. To be self-aware is inside-out – to master the inner world.