I would say that the word "forest" suggests a large area covered with trees in a natural state; whereas the word "woods" suggests a smaller stand of trees which may be either virgin growth or planted. In the case of Robert Frost's poem "Stopping by Woods on a Snowy Evening," I think the word "woods" suggests a small stand of trees, and, since they belong to a man who lives in the village, it would seem that the trees are planted for a purpose. This late in the year, just a couple of days before Christmas, the only trees worth looking at would be fir trees. The deciduous trees that beautify New England in the fall would be bare skeletons. The poem is set in New Hampshire, where Frost lived for many years. The title of one of his collections of poems is New Hampshire. One of the cash crops of this state is Christmas trees. These are shipped to the big cities of Boston and New York. Whereas the deciduous trees would not be pretty in late December, the fir trees with their branches whitened by snow are such a beautiful sight that they often appear on Christmas cards, and they would be worth stopping to look at.
The speaker, no doubt Frost himself, feels a little guilty about stopping there. He says:
Whose woods these are I think I know.
His house is in the village though;
He will not see me stopping here
To watch his woods fill up with snow.
The owner would not understand that anyone might just stop on a cold night to enjoy the beauty of his trees. People in rural New Hampshire do not go to a lot and buy their Christmas trees from a man in a leather jacket who is living temporarily in a littler house-trailer. They go out in the woods and cut down a sapling. The speaker of this poem knows that the owner of the woods would think he was sitting there in his sleigh looking at the trees because he was thinking of chopping one down and taking it home to be decorated. This gives the poem dramatic tension, and Frost has said:
Everything written is as good as it is dramatic. It need not declare itself in form, but it is drama or nothing.
"Stopping by Woods on a Snowy Evening" communicates the speaker's aesthetic enjoyment of the pure beauty of the fir trees with their branches filling up with snow, while at the same time it tells a little dramatic story. The speaker might in fact be tempted to steal one of the saplings for a Christmas tree, but if so he resists the temptation and drives on home. Perhaps he prefers not to desecrate the beautiful picture by leaving one bare stump in the midst of the green and white fir trees.