According to a traditional rule, convince is used
to indicate mental acceptance, and persuade to indicate mental
acceptance followed by action. Thus you convince someone of the
truth of a statement or proposition but persuade someone to do
something. By extension you use convince, but not persuade,
with a that clause. Thus you should say
By convincing me that no good could come of staying, he persuaded me to
leave.If you accept this distinction,
then you should not use convince with an infinitive: He
persuaded (not convinced) me to go. In an earlier
survey, a majority of the Usage Panel upheld this distinction. But the
use of convince with an infinitive has become increasingly common
even among reputable writers. In addition, both persuade and
convince see frequent use with that clauses to indicate the
acceptance of truth: I convinced (or persuaded) the
receptionist that the matter was urgent. Thus, the traditional rule
does not appear to have much of a future.