Persuasive means insiting

Many times they are the same. There is a slight difference. A person might persuade or convince another person to do or believe something. Something (not a person) might convince a person (not persuade). He persuaded me to go on the trip with him. (Could have said "convinced") After seeing the snow on the roads, I was convinced of my choice not to drive. Also, ''convince' has a slightly (not much) more positive sense associated with it than 'persuade'.

A quick check of the etymology tells us that "convince" implies a sense of one person conquering another with their argument, and "persuade" implies that the person urges or leads the other person to believe or accept something. It's a subtle difference, but useful to know as a writer.