I agree with both existing answers (by FumbleFingers and Baz). But I think perhaps I can put the difference in my own words and illustrate it with some examples, so I'm going to post an answer anyway.
I think made of expresses composition, while made from expresses transformation. In many cases, the two express something roughly equivalent:
1. This book is made of paper. (Paper is the material or substance the book is composed of.)
2. This book is made from paper. (Someone took paper and turned it into a book.)
In this case, paper can be interpreted as a constituent material or a source material, so either preposition works. I believe this is true in many cases. In your table example, for instance, wood can be interpreted either way, so either preposition works.
But in other cases, only one or the other is appropriate:
3. *This wine is made of grapes. (Grapes are the material or substance the wine is composed of.)
4. This wine is made from grapes. (Someone took grapes and turned them into wine.)
I don't think example 3 works. I think that in wine form, the grapes have been consumed and are no longer grapes. As a result, there's no compositional relationship between grapes and wine to be expressed. We can express a transformational relationship, though: the grapes became wine.
The following idiom only works with the compositional sense:
5. Let's see what you're made of. (Idiom: let's see what substance you're figuratively composed of.)
6. #Let's see what you're made from. (Let's see what substance was transformed into you.)
I marked example 6 with a # to indicate that it fails to convey the same thing that example 5 conveys, but the sentence would be appropriate in other circumstances--for example, if you were talking to a robot before taking them apart to see what materials they were made from.
I'll present just one more example. The following is taken from COCA:
7. The sample population was made of all 55 female teachers at the two TQA pilot schools in Jerash, Jordan.
8. *The sample population was made from all 55 female teachers at the two TQA pilot schools in Jerash, Jordan.
In example 7, made of clearly indicates composition, not transformation. And when I replace it with made from in example 8, it no longer makes any sense; the teachers didn't undergo a transformation.
Hopefully these examples give you a good sense of the difference between the two.