on Just Publishing Advice:
The adverbs continually and continuously are pretty similar. But they have slightly different meanings.
In many situations, you could probably use either one. It would make very little difference for your readers.
It’s not a classic grammar mistake. However, if you want to make sure you are using the correct word, it helps to know the respective meanings.
In essence, it’s about actions that either never stop or actions that stop and start.