If you convince someone, in the end this person will take up the same opinion. If you persuade someone, you talk the person into something that he or she does not really want. The persuaded person, does not necessarily pick up the same opinion.

Persuade: Means to influencing someon's thoughts or actions. Convince: Means to causing someone to do something by your influential thoughts. Both of them are not the same; There is a pretty distinction between them.