There's no difference. The two words have different origins/ woods = Anglo-Saxon origin forest = French origin

In practical use, I would refer to something as a "forest" if it was quite large (eg the National Forests in the US) and as "the woods" if it was smaller, maybe few acres in size. To add confusion :) if I am out hiking in an area with a lot of trees, even in a lark US National Forest, I would say I'm "out in the woods".