They're very close in meaning, so your best bet is to look at the etymology. This really helps a lot. "Convince" has the meaning of "to win someone over". That is, you have "conquered" them with your argument. "Persuade" has the meaning of "to push someone in a certain direction". This has a more agreeable feel, and doesn't imply that there's a winner/loser relationship. So you see, there is space for overlap. Have a check of the words here: www.etymonline.com ...these are not dictionary definitions of course, but very useful to know.

persuade to do something but convince to believe something generally. After I explained, he was convinced that there is no danger in rock climbing. A good salesman can always persuade the possible buyer to buy.