These words both refer to when a person influences another person to do or believe something:
He persuaded me to move to New York by telling me about how exciting the city was.
He convinced me that New York City was an exciting place to live.
However, there are a few differences. We persuade someone TO do something. Persuade is used for influencing someone to take action. We convince someone THAT something is true. Convince is used for facts or beliefs:
My mother didn't want to go to the doctor, but I persuaded her to make an appointment.
persuade --> action of making an appointment
My mother didn't want to go to the doctor, but I convinced her that it was a good idea.
convince --> fact that seeing the doctor would be a good idea
You can also use the adjective convinced to describe it when a person is completely sure of their belief in something:
I'm convinced that war is always wrong.
Somebody can persuade you to do something... even though you're not convinced it's the best thing to do!
My friend persuaded me to become a vegetarian, but I'm not convinced I'm getting enough protein in my diet.
An argument that is very strong and effective in making you believe something is a convincing argument:
The lawyer presented a convincing argument that his client was innocent.
A person who is very good at influencing you to do things is persuasive:
My father managed to get the local law changed in the town. He can be very persuasive.
Finally, the noun persuasion refers to the act of influencing or encouraging somebody to do something:
After a lot of persuasion from family members, my grandmother finally moved into a retirement home.
It often helps to sever associations between smoking, socializing, and alcohol. Save smoking for solitary sessions. Plan your social life around nonsmoking friends and venues. Cut out alcohol during at least the first two months of quitting; this seems to boost willpower to keep up the good work.