Well, one way of drawing a comparison between these two excellent poems would be to consider how both poems present earthly cares and duties and the speaker's attitude towards them. Note what the speaker in "Birches" says:
I'd like to get away from earth awhile
And then come back to it and begin over.
May no fate willfully misunderstand me
And half grant what I wish and snatch me away
Not to return. Earth's the right place for love:
I don't know where it's likely to go better.
Thus we see that after the speaker desires to get away from earth for a while metaphorically by climbing the birches and swinging on them, the speaker clarifies that he is not wanting a permanent escape, like death. He just wants to experience a temporary break from earthy responsibilities and cares.
We see this theme evident in "Stopping By Woods on a Snowy Evening" as well, as the speaker is entranced by the beauty of the woods and almost overcome by a desire to stay there, enjoying the sight, but reluctantly feels the pull of various commitments dragging him back to reality. Note what the final stanza says:
The woods are lovely, dark, and deep,
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep.
The speaker thus desires the same kind of break and freedom from responsibilities that the speaker in "Birches" does, but even though the woods are described as being so attractive, the speaker reluctantly realises that he has "promises to keep" in the form of various commitments and responsibilities that he cannot dispense with just yet. The repetition of the final line suggests the speaker's weariness at the thought of the "miles" that lay before him.