As summarized by Channie Koplowitz Stein.
The two major personas of parshas Vayigash, Joseph and Judah, are intertwined like the twin wicks of a Havdalah candle. Indeed, each is symbolic of a different approach to serving Hakodosh Boruch Hu. Joseph, in Egypt, 
One of Jacob's main goals in sending Judah ahead was to create a clear separation and demarcation between his family and the Egyptians.
remains the consummate tzadik, never straying from the right path, but keeping his observance secret, hidden within himself, insulated from the spiritual dangers of Egyptian culture. Judah, on the other hand, strays occasionally while associating with the world around him, but admits his errors and returns to the straight path.
When Joseph invites the family to join him in Egypt, Jacob sends Judah ahead to prepare for their arrival. How was Judah to make these preparations? Not only was he to scout out an appropriate neighborhood for the family, separate from the overwhelming influences of the depraved Egyptian culture, but he was also instructed, as Rashi points out, to found a beis medrash, a place where the Torah way of life and the connection with HaShem could be nurtured; a place that would protect their young from the negative influences of a degenerate culture.
One of Jacob's main goals in sending Judah ahead was to create a clear separation and demarcation between his family and the Egyptians around them, between the kedusha of the Jewish way of life and the tumah of the Egyptians. This kind of separation is the exact definition of the word havdalah; and the purpose of the Havdalah ritual at the close of Shabbat is to separate the holy day of Shabbat from the coming mundane days of the week.
But where did this idea and the accompanying ritual originate? The Shevilei Pinchas cites the Gemarah and the medrash and tells us that Adam was the first to perform the Havdalah ritual. Adam had seen that clear, pure light of creation for thirty six hours. Then he sinned and, as night fell at the end of that first Shabbat, Adam was afraid he would be bitten by the serpent, that he would again succumb to the lure of the evil inclination. HaShem, in His infinite mercy, then showed him how to create fire and drive away the serpent. Take two stones and rub them together or, as an alternate explanation, two pieces of wood, and see the sparks of light come out to light the way out of the darkness.
If Man will do the work, HaShem Himself will send down greater fire to illuminate the darkness and help Mankind on his spiritual journey. Adam, with the inspiration HaShem gave him, created the fire that joined with HaShem's fire from above; and Adam blessed the One Who creates the lights of the fire.
But what exactly is this work? When we light the Havdalah candle at the end of Shabbat, we may be leaving the spiritual cocoon HaShem provides for us one day a week, but we light our candle and ask HaShem to send us His light from above to help dispel the darkness of the influence of the yetzer hara', the "evil inclination". In order for us to create light, however, we need the friction of two stones rubbing together. HaShem taught Adam, and continues to teach us, that it is only through the confrontation between two stones, two forces, the forces of good and evil, and harnessing them together, that one can create light. Only by accepting that we have an evil inclination, confronting it, and battling to overcome it with our good inclination, can we grow in our spiritual essence and increase the light of our existence.
Here we have the symbolism of the two stones. One is, indeed, our yetzer hatov, the impulse to remain pure even 
Even when one falls, one has the ability to do teshuvah.
in an impure world. The other stone is our mortal selves, tied to the earth and subject to fall and sin. But even when one falls, one has the ability to do teshuvah, to return and regain the power to produce light. But we must take the initiative and let our evil inclination feel the power of our good inclination, and thereby again bring light to the world.
The two flames of the Havdalah candle represent the original two stones and these two strains in our lives, and the two approaches of Judah and Joseph. Judah succumbed to his passion, but then confessed and repented. His relationship with Tamar was, in the final analysis, the performance of a levirate marriage. Although she was beautiful, and he brought her into his home and raised the two sons she bore him, he never knew her again. He emerged from his fall on a higher spiritual level than he had been on before. Joseph, on the other hand, never fell. He withstood the wiles of Potiphar's wife by keeping the spiritual image of his father's teaching before him at all times. He was pure and remained pure.
We already have a hint of these characteristics from the mothers who bore them, and we will continue to see these characteristics in their descendants. Leah, Judah's mother, was originally destined to marry Esau and be submerged in an alien lifestyle. She cried and expressed her emotion. She overcame her destiny, married Jacob, and become the mother of six of the tribes. She struggled and overcame, and merited to be the mother of kings and priests who must outwardly express their commitment to Torah and Torah values. Perhaps her strength in rejecting the lifestyle that appeared to be her destiny gave Judah the genetic strength to rise above his failings so that he could become the ancestor of kings, involved in the world but rising above it.
While Leah's physical appearance is described only by her "soft eyes", both Rachel and her son Joseph are described as "beautiful in form and beautiful in appearance". Rachel never had the challenge of adjusting to the possibility of life with a degenerate. She remained pure and beautiful throughout. Her inner life was tranquil. Joseph, too, remained pure and untainted, in spite of his physical beauty. He kept his religion within himself, always picturing God at his side, even in Pharaoh's court.
The symbolic rope that binds one to his Maker may become stronger through the knots used to reattach it when it is severed through sin, but the rope that was never cut remains more beautiful. It takes both of these approaches to create the true light. One must recognize when there is a time for silence and when there is a time for speech.
When the brothers return to Joseph with their youngest brother Benjamin, Joseph can longer contain his secret within himself and enters the world of Judah. He blurts out his identity and begins the process of reconciliation and healing in the family. As the Prophet Ezekiel says in the Haftorah of parshas Vayigash, the redemption will come
We hold the two flames of the Havdalah candle together to signify that there is no longer a distinction.when the wood of Joseph and the wood of Judah will become one; when the inner, tranquil, spiritual world and the outer world of spiritual struggle will become one, as both worlds acknowledge the light of the Truth of HaShem and His Torah.
We hold the two flames of the Havdalah candle together to signify that there is no longer a distinction and separation between the two worlds of inner and outer spirit. We have prepared ourselves and our children for the outer world with a safety net of family, Torah education and community so they will not fall off the tightrope as they cross the perilous bridge of life.
As we say the Havdalah prayer and bless the light, we lift our hands to the flame and symbolically bring down the forces of the four angelic realms, of Gabriel, Michoel, Uriel and Raphael. As HaShem makes peace among the disparate angels and forces above, between the fire of the Angel Gabriel and the water of the Angel Michoel, so may He soon create peace in our lower world, so that the lion of Judah and the ox of Joseph will also together sing His praises on earth.