Most of the time these are used interchangeably and nothing turns on it. In a contract, plenty turns on it. "Continuous availability" and "continual availability" describe obligations that differ by every gap in between — and the gap between them has been argued over in front of judges more than once.
An adjective meaning happening repeatedly, with gaps in between.
An adjective meaning happening without any interruption at all.
Quick way to remember
Ask whether the thing stops and starts. If there are gaps, it is "continual." If it never pauses, it is "continuous" — which ends in "-ous," like "one unbroken." In anything you will sign, choose deliberately.