As recorded by Fowler- (p.178)
Continually had been used to mean both 'incessantly,constantly, all the time' and less strictly (OED) with frequent repetition, very frequently'.
Continuously is used when the unbroken nature of a process (in space or time) or set of events (e.g. unbroken terms of office) is being stressed.
- The choice is largely governed by the degree with which continuity is being emphasized.
- I'm afraid you continually make the same mistakes.
Or simply put- when you do something continually, you may take breaks in between, but when you do it continuously, you don't stop at all. Continually is good for describing something you do repeatedly, or on a regular basis. (vocabulary.com)