difference - "Would you mind if I [do something]?" versus "Would you mind me [doing something]?"

Is there any difference between the following sentences?

Would you mind if I smoke a cigarette outside?

Would you mind me smoking a cigarette outside?

Would you mind if I prepare a tea cup for you?

Would you mind me preparing a tea cup for you?

To me, "Would you mind me [doing something]?" and "Would you mind if I [do something]?" seem equivalent ways to ask before doing something, but maybe there are differences between those phrases that a native speaker would catch.

Answer

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..?