May I know, please what the exact difference between the two is?
If a doctor said to me 'If it's no better in a week, come back' I wouldn't expect to be fully fit in a week. I think in that context 'no better' emphasizes the comparative nature of the adjective. 'If you're not better...' sounds to me more final. To convey the same meaning, in that context, as 'If you're no better...' with 'not' the doctor would have to say 'If you're not getting better...'
So in some contexts I think 'no + <adj>' doesn't mean quite the same as 'not + <adj>' . (But maybe this is one of those cases 5jj knows, where a teacher tries to find a difference where there is none. )