“Continuous” means “without ceasing.” “Continual” means “stopping and starting” or “regularly.”
You’ve been breathing continuously since you were born. You’ve been complaining about your college loans continually since 2007.
On a hot day in July, your air conditioner runs continuously, but the bill for the electricity arrives continually, and all summer long.
Politics provides continuous embarrassment. Political campaigns, mercifully, evoke such a feeling only continually.
Here’s how I remember the difference: “continuous” contains an “s,” which is the sound in the word “ceaseless.”
Will knowing this distinction make a big difference in your writing? Not much -- by itself. But as you accumulate this kind of thing, your writing will become more confident and precise, and therefore more persuasive.
The best reason to learn this stuff? It improves not just the way you write. It sharpens the way you think.