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.
Both words describe something that keeps going, but in different ways. Continual means happening again and again, with breaks in between: continual interruptions, continual complaints. Continuous means going on without any break at all: a continuous hum, a continuous line, continuous rainfall for three days.
A simple test is to ask whether there are gaps. If the thing stops and starts, it is continual. If it never stops, it is continuous. A phone that rings every ten minutes is continually ringing. An alarm that never stops is ringing continuously.
In everyday speech the two are often swapped, and in some fixed business phrases, such as continuous improvement, the "ongoing" sense has become standard. In precise writing, especially technical or legal writing, the difference matters.
The meeting was disrupted by continuous interruptions.
✓The meeting was disrupted by continual interruptions.
Interruptions come and go, with gaps between them: continual.
The plant runs on a continual 24-hour cycle.
✓The plant runs on a continuous 24-hour cycle.
It never stops, so continuous.
Draw one continual line without lifting your pen.
✓Draw one continuous line without lifting your pen.
An unbroken line is continuous.
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.
Does the same difference apply to continually and continuously?
Yes. "He continually checks his phone" means over and over. "The pump runs continuously" means without stopping.
Is it "continuous improvement" or "continual improvement"?
Both appear, and continuous improvement is the more common business term. Some quality standards prefer continual, since improvement happens in steps, not without pause.