There is a subtle yet interesting distinction between these two words. They are often used interchangeably, even though that may not grammatically correct.
PERSUADE: to motivate a person to do something
CONVINCE: to lead someone, by argument or evidence to understand or believe something
I convinced my husband that a kitchen renovation was a great idea and then I persuaded him to start dismantling the cabinets the following weekend.
PS I may have used bribery or blackmail to achieve this result, but that is fodder for another blog!