difference between "convince" and "persuade"

[Thunu]

Business English Lesson, Advanced Level

ESL/EFL Test #15 "Advertising", question 5

You have to ......... the public that it is in their own interest to read the instructions on all our medicines.

(a) convict
(b) conviction
(c) convince
(d) convincing

Business English Lesson, Advanced Level

ESL/EFL Test #15 "Advertising", answer 5

You have to convince the public that it is in their own interest to read the instructions on all our medicines.

Correct answer: (c) convince
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Can you explain the difference between "convince" and "persuade", please?

Thank you
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Reach an agreement as ploughing, don't let argument happen at crops.

[Mister Micawber]

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I think that they are essentially synonymous when used. 'Convince' suggests intellectual reasoning while 'persuade' suggests emotional reasoning.
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Native English teacher at Mister Micawber's