These words differ in the shades of meaning. The root of persuade is "sweet talk," and it means to bring someone to the desired point by means of clever argument, entreaty or inducement. The root of convince is " overpower," and it means to make a person unable to maintain a contrary position by force of logic.

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In modern usage, not much. I think they are close synonyms and interchangeable. I'm thinking of sentences that use one and swapping it for the other, and they both work. Google finds me a complaint by someone that "An anonymous copyeditor working on my new book unilaterally changed each usage of 'persuade' to 'convince.' I had to change them all back." So MAYBE there is some doubtful difference that some "experts" in English think is important. MAYBE "persuade" emphases the process, the effort, the attempt, and the person doing the persuasion, and "convince" emphases the person being convinced and the completion of the process. persuade: cause (someone) to believe something, ***especially after a sustained effort*** convince: cause (someone) to believe ***firmly*** in the truth of something MAYBE conviction suggests a deeper and more permanent state. MAYBE you can be persuaded of something all along, and "convince" implies a change in attitude. And maybe not. I see that Dickens, in "David Copperfield, says, both: "I had no gentler feeling towards anyone than towards the pretty creature who had been my playmate, and whom I have always been persuaded, and shall always be persuaded, to my dying day, I then devotedly loved" and "I have always been convinced I am indebted to Miss Betsey for having been born on a Friday." One difference is that there is a noun, "persuasion," for the process of persuading, that is an ordinary word in common use. There isn't anything comparable for "convince." "Conviction" means the opinion that is finally held. "Convincement" exists as a word but it is strange and archaic. If you are reading something before the 1900s there are a number of old-fashioned usages of both words, so give us the context.