They're very similar. In fact I'm having a hard time thinking of an example where they aren't interchangeable. Here are some examples of where you might use them though: "It's hard to adapt to a new city. / It's hard to adjust to living in a new city." "I have to adjust to my new schedule. / I have to adapt to a new schedule." It's a very subtle difference, but I'd say "adjust" is more like "get used to" as in the way you feel about it, and "adapt" is more changing how you do things to accommodate. I think most people would consider them interchangeable though.