These two verbs are too often confused, even by experienced writers. Perhaps the confusion stems from the acceptable use of either in certain contexts. We either convince or persuade someone of the value of a proposition or of the goodness of a certain action. But the two verbs part ways when we try to make them lead into complementary infinitives with to...it works with persuade but does not with convince.
We can convince a person that a statement is correct; we can convince him of its correctness; but we cannot convince (though we can persuade) him to believe it or to act on the belief. In all the examples here, the use of convince is incorrect:
All these examples should use persuade (or prevail upon) where they use convince.