The difference is worth understanding.
In a recent slate article, K.C. Cole writes:
Persuasion requires understanding. Coercion requires only power. We usually equate coercion with obvious force, but sometimes it’s far more subtle. If you want people to stop smoking, for example, you don’t need to make it illegal; you can simply make smoking expensive (raise taxes) or offer bribes (lower health insurance premiums). Both are still coercive in that the power to give or take away resides entirely in the hands of the “coercer.”
Persuasion is fundamentally different because it relies on understanding what smoking does to the human body. Someone who’s persuaded of its dangers has an incentive to stop that’s entirely independent of anyone else’s actions.
I agree that coercion involves the use of (or the threat of) force.
Where I disagree — and where this gets slightly murky — is that I don't think you need to fully understand something (at least at a conscious level) to be persuaded to act. That assumes persuasion is rational.
I think you are persuaded by appeals to the irrational — emotions, psychology, and imagination.
Understanding something (e.g., what smoking does to the human body) largely comes from facts or arguments that appeal to intellect. When I get you to do something based on facts and reason I'm convincing you to act, which is different from persuading you to act.
Seth Goldin devised an interesting heuristic to think about this — “Engineers convince. Marketers persuade.”
It’s a distinction I think about often in teaching. If I get students to do things a certain way for fear of getting an F or hopes of getting an A, it means I’ve influenced their behavior for the duration of the class. If I’ve managed to persuade them that my method has merit, I’ve likely made converts for life.
Cole argues that you can be coerced into doing something for the duration of class, yet persuaded by merit to do it for life. That's an appealing argument but it's flawed.
If you're persuading someone to do something by merit then you're appealing to intellect and reason not emotions or imagination — that's not persuading them, it's convincing them.
While morally better than coercion, I doubt Cole's appeal to reason alone would create a lifelong change in his students. Such a successful outcome (changing behavior for life) would likely be the result of a confluence of factors, not just one.
If I'm trying to get you to do something, there are a number of possible end states (for simplicity, I'll remove coercion). You can be (1) convinced of something but don't take action (e.g., I can convince you that smoking is bad for you, yet you fail to quit); (2) convinced of something and you do take action (e.g., I convince you smoking is bad and you quit); (3) convinced and persuaded (e.g., maybe you were in the camp of #1 but now I've persuaded to act); (4) unpersuaded and unconvinced; or (5) unconvinced yet persuaded to act.
I think Cole convinced but didn't persuade his students (#2).
I looked up ‘persuade/convince' in my copy of Garner's Modern American Usage. The entry reads:
persuade; convince. In the best usage, one persuades another to do something but convinces another of something.
Of course, coming from a usage dictionary you get also get usage instructions:
Avoid convince to—the phrasing *she convinced him to resign is traditionally viewed as less good than she persuaded him to resign.
But that means that you can never be convinced to do something – only persuaded. I don't agree.
I think Seth Goldin is closer to the mark. He points out:
Persuasion appeals to the emotions and to fear and to the imagination. Convincing requires a spreadsheet or some other rational device.
You can convince someone to do something based on reason. You can coerce someone to do something under threat. The way to persuade someone, however, is to appeal to their emotions.
The hardest thing to do is convince someone they're wrong. If you find yourself in this circumstance, attempt to persuade them.
It's easier to persuade someone if you've convinced them, and it's easier to convince them if you've persuaded them.
Persuading > Convincing > Coercion
Ideally you want to convince and persuade.