I like the answer above that deals with your question in general terms, and I've been hoping a teacher who is more grounded in the term "approach" than I am would answer for you. It's been two hours, though, and no one has, so here goes. I think you are asking about these terms as they pertain to education, so that's the direction I will take.
Approach is the way in which you will approach the piece of literature you are teaching. You may center on the diction, or the theme, or the structure, or the romantic (or modern, etc.) nature of the piece; whatever you choose to teach. Approach is what you are going to teach.
Method refers to how you are going to teach it: lecture, small-group work, PowerPoint, worksheet, class discussion, etc.
In a sense, then, as mentioned above by the previous editor, in education, too, approach is general and method is specific.