"Made of" or "made from": what is the difference?

This question came up the other day in class, what is the difference between made of and made from. If something keeps its form, we use made of and if the form is changed we use made from.

For example:
Tables are usually made of wood.
We can still see the wood.
Here is another example:
Omochi is made of rice.


On the contrary,
Sake is made from rice.
The rice used in sake is no no longer rice, it has changed into sake.

So made of is used if the material of which the thing is made can still be seen and made from if the material can no longer be seen.