The adverbial forms, continually and continuously, are often interchangeable.
She sniffed continually / continuously all the way through the film and disturbed everyone around her.
But when the meaning is clearly very often, rather than without a break, continually is preferred:
I've got a very bad stomach upset and I'm continually running to the loo.
Here, continually is behaving as an adverb of frequency, cf. always, all the time, constantly. If we arranged such adverbs along a continuum of frequency, starting with least often and ending with most often, it would read:
never > rarely > occasionally > sometimes > often >generally > nearly always > constantly/continually