Hi!
"To convince" someone of something means that you'll have to present facts, logic, arguments, so that the person will believe the truth of something. In short, when you are convinced of something, you "overcome" or "conquer" your previous beliefs (and this is the meaning of the original Latin word).
It is sometimes said that a person is convinced by reason or logic, but is persuaded by using its feelings or emotions, but this is not always true, so that nowadays one often uses the two verbs interchangeably. If you mix them, that's not a big deal :)
Still, you have to remember the construction that is different for each verb :
You persuade someone to do something.
You convince someone of something.
Have a nice day!