This is one that I see being confused all the time; it’s not too difficult to distinguish the two, although I’d love to know if anyone has a special little rule they use to do so. In both, stuff keeps on happening, but the time the stuff spends happening is either broken or unbroken. So …
Continual – constantly or frequently occurring. “There are continual alarms throughout the day – it drives us mad” – but the alarms stop and start
Continuous – without interruption, unbroken. “A continuous alarm indicates a bomb threat and all areas must be evacuated.”
The most frequent confusion is when things are being discussed that do go on and on, stretching back into the past and forward into the future, like safety procedure updates or quality improvement. I’d be inclined to use continual for this, as by their very nature, they’re likely to stop and start, and are indeed stopped and started by human endeavour, rather than rolling on automatically and seamlessly forever.
You can find more troublesome pairs here.