These can often be used interchangeably: I persuaded (or convinced) him that we needed a new car; He was convinced (or persuaded) of the need for a new car. However, convince means specifically 'to persuade someone to believe something', so it should not be used with a verb referring to an action: I convinced him that he should buy a new car (I made him believe that he should), but: I persuaded him to buy a new car.