The way I have tended to think of social conflict all revolves around perception.  When two people have opposing goals and want to use social norms or concepts of appropriate behavior in order to get what they want, then we have social conflict.

It's not as much about persuading someone as it is about leaving them no socially acceptable option except to agree.  A character could always refuse to do something, they just can't do so doing without looking rude/inappropriate/offensive without winning the social conflict. 

Generally, my rule of thumb is this:
-If you're trying to persuade someone to do something then you're just making skill-rolls against a set difficulty.
-If you're trying to use social pressure (or what other people will think about what's happening) as a way to get someone to do what you want, then you're engaged in social conflict with them.

Social conflict almost always requires an audience who would have a positive or negative reaction to how the conflicting parties behave.  That audience isn't always physically present, but the idea that word of your behavior might get back to them is what makes the conflict a social one.

Examples:
A political candidate is being questioned by a reporter and asked to comment on a potentially scandalous story.  The reporter wants the politician to talk about the story, and the politician wants to avoid it without looking shifty or dishonest.  The reporter likely initiates the social conflict, trying to get the politician flustered enough that she has to answer the question or look bad.  The politician could respond with her own social attack on the reporter by trying to make the reporter's interest in the subject seem inappropriate, or by attempting to throw the reporter off balance by asking about a recent mistake the reporter has made.  Both are trying to get the other to do something (or stop doing something) by deliberately making it difficult for them to resist without appearing rude or inappropriate. 
(Mechanics)
The reporter asks the politician a very pointed question with evidence the politician didn't know about (the reporter probably just tagged an aspect she'd dug up earlier)  This results in a 4 stress hit to the politician forcing her to take the minor consequence "blind-sided" to stay in the conflict.  The politician fires back with an attack on the reporter's credibility doing one shift of stress.  The reporter tries to follow up by pointing out the politician's evasiveness, but rolls terribly, and even with the tag only does one point of stress.  The politician finally claims that we need to "elevate our discourse and transcend the base scandal-mongering of a press gone wild," and with a fate point and a good roll does a 5 stress hit to the reporter which takes her out.  The reporter looks over-zealous and petty for asking the question and fades into the background as people pay more attention to the saintly politician who can now avoid that area of questioning entirely without looking evasive or shifty.

A careless wizard stumbles into the domain of a hostile Lord of Fairy and his court.  The wizard wants to escape unharmed, the Fae Lord wants to kill the wizard.  If the wizard tries to prevent the Fae Lord from killing him by arguing that it would be rude or would look bad to Fairy in general then the wizard has opened a social conflict.  The Fae Lord's objective is NOT to convince the wizard of anything, but rather to convince the audience that his actions would be socially appropriate, or to get the wizard so flustered (by depleting his social stress) that he behaves inappropriately and the Fae Lord can then kill him for that.  Eventually the wizard is successful enough that he runs the Fae Lord out of social stress, and the Lord has to concede and grudgingly allow the wizard to leave unharmed if he wishes to avoid looking like a bad host.

A police officer is called before a review board after she made a questionable decision.  One of the members of the board is clearly out to get her and is trying to get her to lose her cool or admit to making a mistake.  The officer is trying to get the board member to overplay his hand and look petty and vengeful so that the rest of the board will ignore his questions.  The audience in this case is the rest of the review board, the officer and the vengeful board member are the two in conflict.  Eventually the board member wears her down and she is forced to either admit her mistake, or look insubordinate.  She opts to concede the conflict and admit her mistake rather than being taken out and yelling at him.

A nurse is trying to get her patient to take his medication, but the patient doesn't want to because of the side effect.  This is probably NOT a social conflict, it's just a skill challenge (the Nurse's Empathy perhaps, with a difficulty of the patient's Discipline, or possibly a set difficulty).  If instead the patient knew his family was outside in the lobby and he wanted to refuse the request without seeming rude or ungrateful then we may have a social conflict.

Lots of rambling from me, hope it's helpful