I think that you can argue that the speaker is not happy with his decision. The only evidence for that, though, is that he will be telling the story "with a sigh."
But I think that he is content. I think that the sigh is perhaps a nostalgic sigh. It is a sigh of happiness and of remembering. I say this because the general tone of the poem is, to me, not unhappy.
As to the difference, I do not think that there really is a difference. The poem makes quite clear that the two roads were essentially the same. The difference only exists because the speaker has convinced himself that it exists. So I would say that the difference is only real in the mind of the speaker -- he is happy because he feels that he has acted as an individual and a nonconformist.
Since Robert Frost himself is quoted as saying that he wrote "The Road Not Taken" as a playful satire of his friend and fellow-poet, Edward Thomas, the speaker is, at least, not content. Frost would go on treks with Thomas, who wrestled with the decision of which trail to take, anxious that the one that they did not choose would maybe have more flora and fauna that they could have enjoyed.
The sigh of the speaker, who is Thomas, is Frost's mocking of his seriousness in worrying about which trail to take for a simple walk. For, the reader is told that the differences between the trails are negligible:
Then took the other, as just as fair;
And having perhaps the better claim...
Though as for that the passing there
Had worn them really about the same
And both that morning equally lay....
The speaker is simply one of those personality-types that struggles with decisions for his evidence is an illusion, reducing the decision to a negligible importance, if any.
Many people take "The Road Not Taken" as a metaphor for making tough decisions. Two roads in a wood that are essentially the same - but one is less traveled, meaning, it is not the popular choice.
Put this in the context of making this kind of a choice. When you have decided to go the road most traveled, for instance, in the end, has it been as fulfilling as when you decided to go against the popular decision and do your own thing? For some, the answer is a resounding "No." Going with the crowd, for many, has simply resulted in a complacent disappointment. Be honest. How does it feel to be part of the status quo?
On the other hand, taking the road less traveled is often lonelier, more difficult, and sometimes, results in not many people noticing that you went a different way. In my experience, the results of taking a less traveled road have been to my personal benefit, but most often I think I'm the only one who understands the "difference" it made. Perhaps this is why the speaker says this with a "sigh." It did make a difference, a positive difference even, but alas, only he (the speaker) can understand it.