The speakers in both poems are nameless men. The speaker in "Woods" might be richer than the other since he is driving a horse while the other man is a farm laborer. But that's not for sure.
The situation in "Woods" is much more restful -- he's driving somewhere, but he is not doing hard manual labor like the speaker in "Apple Picking."
The themes are different as well. "Woods" seems to be saying that we have to do what the world expects us to do -- we have to do our duty. "Apple Picking" implies that maybe doing our duty is not all that rewarding and we should focus on other things.
This is just to add to the answer already posted.
The two Frost poems you are talking about have a common theme in terms of the way they both throw light upon the interplay of life and death. But beyond this very general philosophical link, while Stopping by the Woods is about the need to go on and complete the incomplete tasks and to live up to the commitments of life before the final sleep, After Apple Picking is more about the closure and an onset of the dead dark, beckoning the final sleep. It is a counter-point to Stopping in the sense that it talks about the impossibility of doing all, accomplishing it all. There would always be some apples left unpicked on the branches of the apple tree. There will always be that remainder.
Other than that, the dream-mode of Apple Picking is absent in Stopping. There are some imagistic and metaphorical similarities in the sleep-death analogue or the image of trees and horses. The drinking-trough appears in Apple Picking whereas we have the horse himself in Stopping. The mood of a stillness and a sort of philosophical melancholy is common to both poems.
please disregard the LISTENING AND SPEAKING portion