from = the material changed of = out of = the material not changed with = indicated tool or effort

There's not a huge difference, generally. There could be some cases where one is better or provides a more precise meaning. 'To be made of' is very general, and it could refer to what material is used in the construction of a product. This version could also be used to describe more abstract personal character traits...like: 'he is made of the right stuff,' implies he has a good, strong character. Like Dan alluded to, ‘to be made with,’ is the most vague. The product may only have a very, very small fraction of something in order to meet the definition of ‘made with.’ You'll often see this in marketing language. Things like: made with real ingredients, made with recycled materials, made with sustainably sourced components, made with real juice, etc. Generally, ‘made of,’ ‘made from,’ and ‘made out of’ are used when there is a significant amount of that material present. If something is ‘made of wood’ or ‘made from wood’ you'd probably be able to tell/see that it's made from wood (in most cases, excluding materials that result from a highly industrialized, refined process).

There is often more than one way to express the same idea. I don't think I can verbalize any clear differences in meaning between those phrases. When I read "the table is made of wood," I have a mental flash of looking at the table and seeing wood grain. When I read "the table is made from wood" or "made out of wood," I have a mental flash of wooden boards and a carpenter sawing and nailing them. That is, they make me think of the origin of the material as well as the material itself. "Be made from" and "be made with" sound a little like marketing or advertising language to me. They suggest that something is _partially_ made of some material. In the case of food, it suggests that some ingredient is included, among many other ingredients. The snack food, Cheetos are made with real cheese," although the main ingredient is corn meal. A Scottish soft drink, Irn Bru, used to advertise "Made from girders," although the only "iron" in the ingredients list is 0.002% ammonium ferric citrate.