When should "persuade" be preferred over "convince", and vice versa?
Let's put some actors on a stage to help us think about the shades of meaning between these words.
Scene: Julie and Robert are discussing the difficulties of a life together. Julie's suitcase is packed and on the floor by her left hand. Their discourse has centered on emotional difficulties as well as logistical difficulties of staying together. As she picks up her suitcase and walks toward the door, Robert says:
If Robert uses "convince," then it seems to me that he is asking if there is a logical argument that would perhaps address the logistical difficulties in question.
If Robert uses "persuade," then it seems to have more of an appeal to Julie's emotions. (There may be a coercive component to "persuade" that is less present in "convince." Think film noir where the gangster type uses the word "persuade" as a threat, for example, "I sees dat you needs some persuading," as he smacks his fist into his palm.)
When confronted with issues of word choice, I often find it helpful to consider the associations and connotations a word has in its different forms. A convincing person or argument is one that other people cannot help but agree with, whereas a persuasive person or argument is one that other people find compelling, but which leaves more room (in my mind) for the opportunity to disagree.
If you have been convinced, you have clearly and decisively changed your perspective. If you have been persuaded, then you have chosen to agree with those you previously disagreed with. In the latter case, it may not be so much that you've found an argument you agree with so much as you're indulging someone who has appealed to you, either positively or negatively.
Prefer "convince" when the change of opinion must be definite or to deemphasize the role of whomever did the convincing. Prefer "persuade" to draw attention to either the arguments used or the individuals doing the arguing, to emphasize reluctance on the part of the persuaded, or when you need the word to contain more vowels.