There is a difference between approach and method. An approach is a way of dealing with something or somebody. A method is the process used or the steps taken to deal with an issue or a person.

I am going to use a very current example for people of the Jewish faith. This week, a Jewish holiday begins called Sukkot. One of the requirements of this holiday is to live in the sukkah, which is kind of a temporary structure or dwelling. For some people, this means eating in the sukkah. Some people, in warmer climates, may even sleep in the sukkah. When building a sukkah, there is an approach that must be developed and a method that must be followed.

My approach for building a sukkah is to determine the space where it will be placed. Generally, I want to build it where it will receive some shelter from the wind, which is usually close to our house. Then I need to determine what materials I will need to build it. After I have determined these things, I need to decide where the lights will go and how it will be decorated. I also must be sure I have a proper roof-like covering.

After I have determined all of these things, then I will follow the method of constructing the sukkah. First, I will build the base. After that, I need to build the walls. Once that is done, I will place poles over the top of the sukkah, so I can get the roof constructed. In my case, I will use a bamboo mat as the roof. After that, I will place decorations in the sukkah and decide where the lights should go so we can see at night. This is the method that needs to be followed when building the sukkah.

The approach is when I figure out all the logistics and the materials that are needed. The method of building a sukkah is the steps I follow to actually build it. This very joyous Jewish holiday lasts for seven days.

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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.