As the LDS and Biblical Christians celebrate Christmas, is it the birth of the same Jesus? It depends on how important you consider the differences to be. When I get asked this question, I find it hard to give a short answer.
For those who believe in the triune God of historic Christianity, the manger is where the Incarnation takes place: the God of heaven becomes and is born as a human child, without ceasing to be the God who also reigns on the throne of the universe. He is the only child who ever existed before he was born. In his flesh, he was conceived by a miraculous act of the Holy Spirit, without the seed of any human male. The mission for which he was born was to live the life we should have lived, and to die the death we should have died.
The Mormon Jesus starts out as a spirit child in the Preexistence, produced by Heavenly Father and a heavenly mother, to be followed by the births of Lucifer and the billions of human spirit children begotten by them since then. The future role of Savior is then offered to Jesus and also to Lucifer, to be decided by who offers a better plan of salvation. Jesus wins the contest, Lucifer is denied the privilege of having a physical body, and Jesus proceeds to receive his.
Already, this Jesus in the manger is not the “only begotten Son of the Father.” (John 1:14)
That's where it gets even more complicated. Spirit children are given bodies through the biological process of human conception. Unlike the rest of us, Jesus gets his body from Heavenly Father himself, who is an exalted human being who still has a now glorified human body. As an exalted human with a physical body, Heavenly Father pre-empts the marriage of Joseph and Mary temporarily to mate with Mary to conceive the human Jesus.
Now, the manger scene is starting to look very different. Mary is no longer a virgin.
The above reconstruction comes from the prophet Brigham Young and from Orson Pratt, one of his apostles. Brigham preached the puzzling theory that Heavenly Father is actually Adam (today’s LDS have disavowed this particular teaching from their prophet). Orson Pratt speculated as to whether Heavenly Father gave Mary to Joseph only temporarily, and would take her back as one of his wives in the resurrection. Both of these leaders deny categorically that Jesus was conceived by the Holy Ghost. Otherwise, they said it would be dangerous to lay hands on women to bestow the Holy Ghost on them, lest they likewise conceive unplanned children.
There is one more, huge problem. At what point does Jesus become a god? In historic Christianity, he is born already God in the flesh, the same God who simultaneously sits on heaven’s throne. Our Jesus is co-eternal with the Father, and was always God. But in LDS teaching, Jesus must start out human, and then be exalted to godhood. (Although both Brigham Young and Orson Pratt taught that we are all gods in embryo.) In 1838, Joseph Smith was teaching that Father and Son are two glorified bodies of flesh and bone, both of them distinct Gods rather than a tri-unity. So when does Jesus get exalted to godhood? It is not clear, but probably not until sometime after his birth in Bethlehem.
Our LDS friends might object that I have distorted the picture. The message they proclaim is that Jesus was born to be the Light of the world. To some extent, we can agree, although we do not agree on who exactly he was. But there has to be more to the story. By itself, the fact that Jesus is the Light of the world comes as bad news if we recognize that Jesus is an impossible act to follow. He is the one truly good kid that makes us all look bad by comparison. If all that Jesus came to give us was the highest example, we are toast. The historic Christian Jesus came to give us far more. He came to take away all of our sin and put us right with God forever, which the LDS Jesus never came to do.
The average Latter-day Saint does not necessarily believe all that their prophets teach about the Jesus that was born in Bethlehem. Many Latter-day Saints may be celebrating the same story told in the historic Christian church. But sooner or later, the truth will catch up to them – hopefully, not at the throne of judgment.
Wishing you a joyous celebration of the Incarnation!