Both are correct. There is no difference in meaning. "Do you mind me leaving.... " is more common. This is a gerund with a specified subject. https://en.wikipedia.org/wiki/Gerund#"Gerund"_clauses_with_a_specified_subject

Do you mind MY leaving the room? (Using the possessive pronoun + the gerund is the correct form. This would be the correct answer on an English language exam.) Do you mind ME leaving the room? (This form is used in colloquial spoken English. These days, it's not considered incorrect to use informally. Languages are constantly evolving, so maybe one day this form will also be considered correct on an English exam.)