Learn English – “know of X” versus “know about X”

Yes, there is a very subtle difference between the two forms

Would you mind if I smoke a cigarette outside?

is equivalent to

Would you mind my smoking a cigarette outside?

since both phrases refer to the action of smoking. In the second form, when referring to the action of smoking and you want to use the verbal noun (gerund) smoking, you have to use my.

Would you mind me smoking a cigarette outside?

refers to me personally and not the action of smoking.

Would you mind if..? or Do you mind if..? is just another way of saying May I..?

I'm trying to think of a context where there is a difference, and it's very hard to do that. What makes this tricky is that the two are pretty much interchangeable in many contexts:

When I think about/think of my mother, who passed away three years ago, I get very sad.

but there are other occasions when one would be slightly (or clearly) favored over the other:

I get excited when I think about football season getting underway.

I can think of 10 reasons why you shouldn't text and drive.

And then there's this gem:

I can't think of it right now, but maybe I will, if I think about it long enough.

If I had to summarize, I'd say that think of seems to mean recalling something specific, while think about seems to mean considering some subject in a more vague or general way for some length of time. The two are sometimes interchangable because there are many situations where you can't think about something without also thinking of it, such as when you are fondly remembering your deceased mother.

^{This is a great question; I hope others weigh in. I think I might be getting close to something here, but I'm not convinced I've nailed it.}