Basically, it's whether the word is followed by an action or an idea. You convince someone to do something. That's an action. You convince someone of something. That's an idea. Check these examples: I tried to convince her (to believe) that I could fly. -You don't need to say "to believe" because it's understood, but the action is there. I tried to convince her of my ability to fly. -This also shows that it can be awkward to use one over the other sometimes. The first sentence sounds much more natural than the second.

convince 1 verb : If someone or something CONVINCES YOU OF something, they make you believe that it is true or that it exists. For instance: Although I soon convinced him of my innocence, I think he still has serious doubts about my sanity. The waste disposal industry is finding it difficult to convince the public that its operations are safe. 2 verb: If someone or something CONVINCES YOU TO DO something, they persuade you to do it. For instance: That weekend in Plattsburgh, he convinced her to go ahead and marry him. see also: http://dictionary.reverso.net/english-cobuild/to%20convince%20sb%20of%20sth