The Church of Jesus Christ of Later-day Saints is what most people think of when they hear the term Mormon, but the term Mormon itself is inaccurate for the religious movement. When Joseph Smith received the revelation behind his Palmyra home that would later become the sacred grove. The revelation that the future prophet received from God and Jesus Christ gave Joseph Smith the divine mission of restoring the gospel to the Earth.
All Mormon groups, even if they do not like to refer themselves as Mormon, share a belief in the necessity for Joseph Smith to restore the gospel on the Earth. Because of this belief, the groups, including the Church of Jesus Christ of Latter-day Saints and the second largest Restorationist group, the Community of Christ, formerly the Reorganized Church of Jesus Christ of Latter-day Saints.
The Church of Jesus Christ of Latter-day Saints and the Community of Christ share the same early history and honor Joseph Smith as the central figure and the person who restored the gospel to the Earth, which had been in a state of apostasy since shortly after the death of Jesus Christ. The break-off occured after the death of Joseph Smith and Brigham Young's rise to power.
Many Mormons chose to follow Brigham Young out west when he made the decision after Joseph Smith was martyred while the first Mormon prophet was imprisoned at Carthage, Illinois. Some Mormons who remained in Nauvoo, including the wife of the late first Mormon prophet, thought that the leadership of the church should pass to the prophet's oldest son.
The Restorationist Faction that did not follow Brigham Young remained in Nauvoo and believed that the leadership of the church should pass to Joseph Smith III. Joseph Smith III was reluctant to assume the mantle of leadership, and the Reorganized Church of Jesus Christ of Latter-day Saints would not be formed until 1866. The early Mormons who did not believe in the prophetic powers of Brigham Young needed to wait for the son of the Mormon founder to have a revelation that it was his calling to be the prophet of the newly reformed church.
The scriptures used are slightly different than those used by the LDS Church. Because the Community of Christ maintained the rights to the Joseph Smith Translation of the Bible, it uses that in its services. Quotes from the translation are found in the King James Version used by the Church of Jesus Christ of Latter-day Saints, and members of th Community of Christ do not have to profess a belief in the historicity of the Book of Mormon. The Pearl of Great Price is not part of the Community of Christ's canon. The chapters and verses in the scriptures used are different.
Other key differences are that the temple rituals which are practiced by the LDS church are not used by the Community of Christ, which simply makes the standard church services available in their temples. The few temples owned and operated by the Community of Christ are open to everyone.
The membership of the Community of Christ does not believe that the Word of Wisdom is a commandment, as the text of Section 89 or the Doctrin and Covenants specifically says it's not. The key difference between the two groups is how each group treats its history. The LDS Church prefers it to be faith promoting, while the Community of Christ now accepts and states that Joseph Smith, Jr., practiced polygamy.
Resources:
Community of Christ History Page
Community of Christ – About Us