In the poem 'Stopping by Woods on a Snowy Evening' Robert Frost characteristically uses the term 'woods.' He could have used the word 'forest' but this term wouldn't have worked as well for his purposes. He often used the word 'woods' - you can look at the poem 'Desert Places' for another example. It's possible that he used the word 'woods' because it has an olde-english quaint connotation. Woods are characteristic features of the English countryside which has had its native forests coppiced into more manageable slices as the land was cleared. They tend to cover less area and to be managed, for example for charcoal or game shooting. The New England countryside is said to be reminiscent of this landscape. Forest can be enormous by comparison, and are often still native trees, or planted up for commercial purposes.
The word 'woods' is of course mono-syllabic, which has impilcations for rhythm and metre - you might want to look at that also. Forests has two syllables.