Forest is a bit more formal and carries the sense of something larger than the phrase "the woods" does. If you told me you got lost in the woods behind your house, I'd imagine some acres worth of trees. If you told me you got lost in a forest, I'd picture a much larger area. It's also worth noting that it is always "the" woods. There is no such phrase as "in a wood" or "wood" because then you're talking about pieces of wood.

Hello Vera I think that the tree is made from wood, and the forest is a collection of tree on a region, for example: the Zirahuen's forest. :)

wood is bigger

In my opinion, "wood" is more often used to name the material (This table is made of wood) and "forest" to describe the area which is covered with trees (Animals live in the forest)

You can say: "Animals living in the woods"; or "Animals living in the (/a) forest" You can't say "Animals living in a woods"; instead, you would say "Animals living in a woodland area" - better to use forest if you use "a". Otherwise, there is no serious difference. (I think) woods is a more British term, but living in Australia, I could use both quite comfortably. :) I think Daisy might be right in that "forest" might be a little bit better in a formal, written piece.