Definitions: 

Continual: Referring to a series of continuous events punctuated by breaks. Continuous: An action that never stops.

When is Continual Used ?

Continual refers to a continuous series of events that have breaks in between. There are several pauses and starts in the action. For instance, It was continually raining today. In this case, it rained for a period of time, stopped for a period of time, and then rained again at uncertain intervals.

Examples of Continual

They were talking to each other continually.Sam continually drove until he reached Texas.She has a habit of studying continually.

When is  Continuous Used?

Continuous or its adverb continuously means a nonstop action. The process goes on without any breaks or stops. For instance, The teachers saw continuous improvement in her academic performance. It means her performance has been constantly good without any lags or interruptions.

Examples of Continous 

He runs continuously for 45 minutes every day.She snacks continuously while watching the television.Over the last 5 years, the business has grown continuously.

Examples in Academic Writing

Incorrect: We had to mix the reactive compounds continuously to avoid explosion. Correct: We had to mix the reactive compounds continually to avoid explosion.