Cameryn, that is a great question because the question itself really brings to light some of the most important truths about the LDS church. 
 
First, you ask when and why we split from "the Christian Church". My question to you would be what is "the Christian Church"? This was the very question that a 14-year old boy named Joseph Smith struggled with in upstate New York back in 1820. 
 
Joseph grew up in a strong Christian family and he had tremendous faith in Jesus Christ. Joseph had a strong desire to unite himself with the Christian Church, however, as he looked around, he saw many churches: Catholic, Methodist, Presbyterian, Baptist, etc. Each of these churches professed to be the Christ's Church however they each differed in many points of doctrine and argued constantly with each other about what the Bible said. As Joseph attended these various churches and searched the Bible for the truth, he quickly came to the realization that there was no way he, a 14-year old boy, could come to any reliable conclusion simply by reading the Bible and listening to these preachers. There were just too many different ways one could interpret the various passages in the Bible. 
 
Joseph described his predicament like this... "In the midst of this war of words and tumult of opinions, I often said to myself: What is to be done? Who of all these parties are right; or, are they all wrong together? If any one of them be right, which is it, and how shall I know it?" 
 
While Joseph struggled with this problem, he came across the scripture in James 1:5 which read: "If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him." 
 
Joseph came to the realization that the only hope he could ever have in resolving this question would be to ask God himself in prayer which of the churches he should join. One morning, Joseph went in to a grove of trees near his home and knelt down to pray in order to receive an answer to this question. As he prayed, a vision opened up to him. These are his words describing the experience:  
 
"I saw a pillar of light exactly over my head, above the brightness of the sun, which descended gradually until it fell upon me... When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other—This is My Beloved Son. Hear Him! 
 
My object in going to inquire of the Lord was to know which of all the sects was right, that I might know which to join. No sooner, therefore, did I get possession of myself, so as to be able to speak, than I asked the Personages who stood above me in the light, which of all the sects was right (for at this time it had never entered into my heart that all were wrong)—and which I should join. 
 
I was answered that I must join none of them, for they were all wrong; and the Personage who addressed me said that all their creeds were an abomination in his sight; that those professors were all corrupt; that: “they draw near to me with their lips, but their hearts are far from me, they teach for doctrines the commandments of men, having a form of godliness, but they deny the power thereof.” 
 
 
Through this and other revelations that Joseph received, he was called as a prophet of God and given authority to restore the true Church of Jesus Christ upon the earth. The church that was restored through Joseph was not a spin-off or reformation of any existing church on the earth, rather it was a complete restoration of the Church that Christ had established during His mortal ministry but was lost from the earth with the rejection and death of His original apostles. The keys and authority to re-establish and direct Christ's church on the earth were given directly to Joseph Smith by John the Baptist, Peter, James and John and other heavenly visitors. This is one of the principle differences between the LDS church and all of the other Christian denominations out there. 
 
So, to answer your question "when and why did the Latter Day Saints split from the Christian Church?", the answer is simply "Never". The Church of Jesus Christ *is* the original church that Christ himself established, both during the time of his mortal ministry as well as in these latter days.

According to The beliefs of the Church of Jesus Christ of Latter Day-Saints it was Christianity that fell away from Christ and His Apostles and the LDS Church is the Restored Church of Jesus Christ. 
 
We are the fullness and the truth of the Gospel of Jesus Christ restored to the Earth through the Prophet Joseph Smith. So in our faith we never separated from Christianity. We believe that Christianity separated itself from Christ by allowing the Prophets and Apostles to be persecuted and martyred. When the Apostleship was lost to the world it entered a period of spiritual darkness called the Great Apostasy. During this time men did there best to preserver the doctrine and Gospel of Jesus Christ but without the Apostleship to receive revelation from Heaven to Guide the Church it slowly but surely became filled with man made doctrine not the true doctrine of Jesus Christ.  
 
There was a need for a restoration to the truth of the Gospel of Jesus Christ, and a need for another Prophet to be called when the time was right to restore the Gospel to the world.  
 
In 1820 a young boy named Joseph Smith Jr. live in Palmyra, New York. In his area was many religious revivals and he was very confused because the various preachers taught the same scriptures with some any different interpretations to destroy all faith in being able to turn to the Bible for answers since it was so easy to interpret the scriptures to fit what the preacher wanted to present.  
 
Desiring to unite himself with the Church that represented God and obtain salvation for his soul he searched the scriptures diligently . One day while studying the Bible he found James Chapter 1 Verse 5 which says if any man lacks wisdom let him ask of God. The prophet stated that this verse impressed upon him like no other verse ever had to mankind before.  
 
He knew after much consideration that the only way to find the truth was a direct petition to Heavenly Father per the instructions given in James 1:5. He determined that he would seclude himself in a nearby grove and pray to Heavenly Father for the answer of what Church to join. In April of 1820 he did this and was visited by Heavenly Father and Jesus Christ in the Grove.  
 
Jesus Christ told the Prophet to join no Church because there was no Church upon the face of the earth at that time that was His Church. They all had doctrine that were not of Christ but of man. He promised that if the Prophet would prepare himself that he would be the instrument in the Lord's hand to restore the True Church of Jesus Christ to the face of the Earth.  
 
Over the next ten year the boy Prophet was visited by other heavenly messengers and prepared to restore the Fullnes of the Gospel of Jesus Christ and His Church. This restoration was completed on April 6, 1830 when the Church was restored in upstate New York.  
 
To get a more full account of the restoration you can read Joseph Smith - History in the Pearl of Great Price which is a book of Scriptures for the members of the LDS Church. The Pearl of Great Price can be found online at http://lds.org/scriptures?lang=eng