Both adjectives describe duration.
Continuous indicates duration without interruption.
Ex. The continuous humming of the fluorescent lights gave him a headache.
Continual indicates duration that continues over a long period of time, but with intervals of interruption.
Ex. The continual street repair disrupted traffic for nearly two years.
The adverbs continuously and continually preserve the same distinction:
Ex. The child screamed continuously as long as its mother was on the telephone.
In this part of the country it rains continually during April.
Constantly means going on and on without change.