Simplest example would be when you are getting a coffee and ask for a paper cup. It seems that both 紙の and 紙で are acceptable. What are the differences?
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My (non-native) intuition here is that by using の you are picking an attribute of the cup, which can really be any salient attribute it has. Syntactically, you are just omitting the noun after the particle. By using で, on the other hand, you are choosing among a given set of options. I most frequently hear this kind of で when picking a meal size (for example 小, 中 or 大).
Look at it this way - you can also answer this question by saying 紙のカップで, but not 紙のカップの.