I finally decided to put my OED CD to good use. Here’s is what I found about historical usage of "convince" vs. "persuade."
I discovered that there is a distinct difference in historical usage between "persuade" and "convince" and that we do not always honor that distinction today. I, for one, am convinced that the difference is worth fighting for. These words help us to discriminate among subtle meanings. St. Paul in the Bible (Ephesisans, Romans, etc.) carefully distinguished among these meanings. They were not used interchangeably, nor are we to translate them from the Greek or Latin willy-nilly.
Now for the evidence:
"Persuade" is from the Latin, persuadere, to bring over ty talking, induce; urge as desirable. Not in general use until the 16th century. What’s interesting is that this word has the connotation of inducing someone to believe something by "inclining his judgement or desire to it." In definition 2b: "In order to persuade, we address the feelings and the imagination. In order to convince, we address the reasoning faculty." (1846, G. F. Graham Eng. Synonyms).
There is in "persuade" a course of action to get someone to believe or have a conviction. "Dr. Fenwick has earnestly persuaded me to try horse-exercise." (1801). There is also the 5th meaning of inducing the doing or practiicing of by argument, entreaty, or the like; to induce or LEAD to by reasoning. "S Augustin persuadeth with him to leaue the Manichees." (1565)
Now, to "convince" means originally from the Latin "to overcome, conquer, convict, demonstrate." L. convinc-ere or con-altogether, wholly + vincere to conquer. So we get meanings like "to overcome (a person) in argument; to prove to be wrong, confute. We are forcing someone to acknowledge a contested position. Or, we are causing a person to admit, as established to his satisfaction, that which is advanced in argument; to bring to acknowledge the truth of; to satisfy or persuade (notice the overlapping) by argument or evidence. To be brought to, or to have, a full conviction; to be firmly persuaded. "I am convinced . . . and have nothing more to object."
(1875)
Ah, but then I came upon this gem under the definition "to induce, prevail upon, persuade." 1958: Another usage . . . that seems to be becoming frequent in Pennsylvannia and New York, is that of ‘convince’ in the sense of ‘persuade’ e.g., ‘She convinced him to clean the cellar.’
So it seems to be reasonable to conclude, from the OED citations, that there is a distinction between "persuade" and "convince" that is historical and worth keeping in our current usage. When we persuade, we are trying to win someone over, via action, to our position, our way of thinking. On the other hand, when we are convinced or convincing, we are suggesting a firm conclusion, an established position, or even the truth. "I am convinced that U.S. landed on the moon." I hold this as firm conviction. I have already been "persuaded" by reasons or emotions to hold this position. I am there, planted firmly on both feet!
Here are two more famous quotes from OED:
1611 Bible Job xxxii 12: "There was none of you that convinced Job, or that answered his words."
1671 Milton P.R. III. "Satan stood . . . confuted and convinced Of his weak arguing and fallacious drift."
Here "to convince" connotes "to conquer." Maybe someone tried to "persuade" Job, but he remained unconvinced. He was not conquered in his beliefs to the contrary.
Likewise, Satan was convinced or won over or at least admitted defeat in his line of reasoning and thinking.
"Convinced" suggests a final position after careful reasoning. "Persuaded" suggests, on the other hand, that we have been manipulated and emotionally coerced to accept a position. The media is amazing in persuading us to believe in anything. They appeal to our emotions, reason, and character to move us toward their proposal.