To answer this question, one first needs to identify the intention of the poet. One should also consider whether he is the speaker or if it is someone else. Is the speaker universal - one who speaks for all of us?
Many interpretations mention Frost's relationship with his writer friend, Edward Thomas, and it is believed that Frost mocks Thomas' indecision in the poem. Be that as it may, it is said that Thomas took the criticism very personally and that this drove him to enlist in the First World War where he was later killed.
The title of the poem "The Road Not Taken" makes it clear what the intention of the poet is: the focus is on the choice that was not made. The prevailing thought is not on the choice that was made, i.e. the road taken. One could also argue that Frost used reverse logic by using this particular title but instead focuses on the other choice - the road that he actually took.
What makes the conundrum more difficult is the fact that the two roads are very similar. There is hardly any difference between the two. The speaker says:
"Though as for the passing there
Had worn them really about the same."
"And both that morning equally lay
In leaves no step had trodden black."
He decides on the other road which was "just as fair" because:
"it was grassy and wanted wear;"
and for this "perhaps" had "a better claim."
Clearly, the speaker is unsure of his choice. He expresses doubt. This uncertainty is emphasized later when the speaker expresses doubt whether he would ever have a chance to tread the other road since the road he takes would lead him onto other, different paths. Furthermore, the pause and repetition of "I" in the penultimate line (a stutter, if you will) asserts this doubt.
It should be patently clear that the speaker was very tentative when confronted with this choice. He expresses his uncertainty throughout the poem. Many interpretations assume that the speaker had not made any choice at all - he was too overwhelmed by uncertainty. Of course, that decision would have been a choice too.
Because of all the uncertainty, it should be clear that the "sigh" would be an expression of the speaker's regret about whether he had made the right choice. This regret is already expressed in the line:
"Oh I kept the first for another day!"
The exclamation mark emphasises the depth of the speaker's anguish for having been so unsure. It is here that the poet clarifies why the title is "The Road Not Taken" and not "The Road Taken." He is tormented by the idea of the possibilities and opportunities that the first road could have held. What if he had taken that road? Did he make the right choice?
The fact that he chose the second road and not the first,
"has made all the difference."
What that difference is we shall never know. The poet himself commented that this poem is "very tricky" and never commented about "the difference" since he, I assume, could not have known either, since an adequate comparison could only have been made if both roads had been traveled. Obviously that did not, and could not, happen.