These two words sound the same and their meanings differ only slightly, which may explain why people regularly confuse them. Here’s my take on the difference between continuously and continually.
Continuously
Continuous indicates a duration without interruption, so continuously means something occurred for a duration uninterrupted.
That whippersnapper played her drums continuously all night long, not even stopping to pee.
Continually
Continual indicates a long period that has interruptions, so continually means something occurred for a long period but may have been interrupted.
The city has been “fixing” this street continually for the entire summer, and they’re still not done.
I hope that helps. Let me know in the comments below if you have tricks for remembering the difference.