logical fallacies

The Appeal to False Authority works because if you can persuade the audience that you or whatever you're citing is a trustworthy source of information, you appear more knowledgeable and therefore more trustworthy yourself. Effectively, it's a way of making arguments seem legit because they were endorsed by a person or group that supposedly has 'the know-how' and 'know-what'.

E.g.

Carl:  Tax cuts for the wealthy are absolutely the way to go! But don't take my word for it. The Harvard-educated, self-made entrepreneur Dick Dickens agrees!

Audience: 

"To be fair, he is an entrepreneur."

"Yeah, and think - he went to Harvard!"

(in this case, what I'm doing is 'blessing the well', by adding irrelevant information to the statement - him being a Harvard alumnus and an entrepreneur isn't relevant. If he's just a businessman that doesn't confer knowledge of economic theory at all, so it's a false authority. However, people get taken in by his achievements and thus trust him more).

The non-sequitur is best paired with a series of priming statements, thing that are likely to resonate with the audience. By associating these agreeable comments with your main point, they'll be duped into thinking they're related and therefore accept the conclusion. It's persuasive - because the points appeared in succession, people will think they're related (post hoc, ergo propter hoc) even though there's no connection between them.

E.g.

Dane: "People have a lot more free time nowadays. They have a lot more free money to spend, too. Therefore this holiday package is a good buy for you, m'am."

You might find yourself thinking, "this is ridiculous, Rationalissimo." And it is! But in real life, the "m'am" (or whoever else is being spoken to) might go, "well, that's kinda reasonable, right?" But the conclusion doesn't follow from the premise. Just because I have free time and money doesn't mean I should spent it on any old things. Otherwise, I'd very soon have exactly no money or time - because I've spent it all on crap.

Because the person is in a position of authority, there may be a perception that what they say has merit even though their position is irrelevant to the topic.

A non sequitur would be persuasive if the person just accepts the argument without proper analysis. The person making the argument probably thinks that they are making a good point, it's not always obvious.  

One thing is that the examples given to explain what a non sequitur is may be exaggerated to help explain, but a real example may appear reasonable. For example you read that there have been numerous attacks by pit bull terriers, therefore your neighbor's pit bull terrier is dangerous. That is a bit less obvious than Bo's example of Buddy Burger serving good food, therefore the owner should run for president. 

Basically if you can identify that it's a non sequitur you won't be persuaded, but if you don't then you may well be.