1. I convinced him that he should do it.
2. I persuaded her that she should do it.
3. I persuaded her to do it.
4. I convinced her to do it.
In all four sentences, 'he' did it.
in #2, for many speakers of BrE, 'he' believed that it was right to do it as a result of my arguments.
In #3 and #4 'he' did it, but did not necessarily believe that it was the right thing to do.
#1 is unacceptable to some speakers of English. Here convince is virtually a synonym of persuade.
p.s. I submitted my response a few seconds after Ron. I'll let mine stand.