persuade - means to suceed in causing someone to do something. it doesn't necessarily mean that the person is convinced.
examples are:
a salesman persuades you to buy something
a marketing agent persuades you to taste their new recipe
you are persuaded by your friend to sign up for a gym class
convince - means to make someone believe (when one is convinced, it is easier to persuade)
you will find in the dictionary/thesaurus that the two words are synonyms (i tried it with MS Word thesaurus, and they are synonyms) but synonyms means the two are the same OR very similar. in this case, the words are similar but not exactly the same