Yes, there is a slight difference, although most people (and even many dictionaries) treat them the same.
Continually means repeatedly, with breaks in between.
Continuously means without interruption, in an unbroken stream.
Ex.: Heidi has to wind the cuckoo clock continually to keep it running continuously. (If it's important to emphasize the distinction, it's probably better to use periodically or intermittently instead of continually to describe something that starts and stops.) The same distinction, by the way, applies to continual and continuous, the adjective forms.
Source: Your Dictionary