To convince someone is to change a person's mind about something. A change in belief or attitude is the primary result, which may lead to a change in behavior. For example, you may believe that a certain car handles poorly. A test drive can change that belief and convince you that it handles well.
Action is usually the focus of persuasion. In general, a person is persuaded to change a behavior. This can seem like convincing because sometimes a change in attitude or belief is needed to inspire a different behavior. For example, a jury may be persuaded to find a defendant innocent. The focus is on causing the jury to make a decision based on a belief, rather than on the belief itself.
Both words can refer to a change in belief. You can persuade someone to consider a new idea, for example. However, strong changes in belief, such as a religious conversion or another fundamental attitude change, are usually associated with the word convince rather than persuade.
The stronger associations of the word convince date back to the origins of the word in the 16th century. To convince a person was to strongly suggest and sometimes force him or her to do something. An older, and now rare, meaning of convince was to overpower or to conquer. Persuade, which also dates back to the 16th century, referred to using speech to influence a person. Convince seems to have been associated with force, while persuade was a softer word, suggesting a change in attitude or belief.