What are the differences between approach and methodology?

They are related, and you can't have one without the other. Your approach is your perspective, how you will be studying the particular concept. There are many possible ways of approaching (studying, analyzing) a topic. Perhaps you want to study the early days of television. You could use a Media Ecology approach (how did the new mass medium of TV interact with and compete with other media, and how did it change the media environment); a Social History approach (how did TV change people's lives?); a technological approach (what changes in the technology improved TV, what problems did early TV have and how did engineers remedy these problems), etc. Depending on your approach, you will then need a way to prove whatever your central thesis is.

This means you will need to do research. Your methodology is important because it is how you intend to prove your points. Once you have a thesis statement, you may decide to defend it using qualitative research (very popular in the Social Sciences-- in the case of a study of TV, you would explore what a specific number of newspapers of that day said about it; or examine what great historians and scholars of media said about it; or interview people who were there in the early days of TV; or compare and contrast the basic beliefs and perceptions that were expressed about TV in its early years, etc). Or, perhaps you will decide upon a more scientific and statistical type of research-- quantitative (this involves deciding upon a hypothesis and then seeking to prove it by using research from taking surveys or hosting focus groups; you would be creating the proper questions and making sure you have a valid sample, analyzing the statistical data from your respondents, and then coming to a conclusion). In a good term paper or even a dissertation, your approach will determine what perspective you intend to use in writing about your topic; and your methodology will provide the units of analysis and the proof needed to back up whatever you are asserting.