Summary: Why is the Book of Mormon not the word of God?
The book of Mormon is a book that claims to tell the history of 3 groups of people from ancient America, two groups, the Nephites and the Lamanites, that came from the city of Jerusalem around 600bc before the Babylonians destroyed the city. The third group, the Jaredites supposedly came from the tower of Babel after God confused the languages. Interwoven in this book is the supposed personal ministry and appearances of Jesus Christ to the inhabitants of America after His resurrection and ascension in ad34.
In September 1823, Joseph Smith was supposedly visited by a heavenly messenger named Moroni. Moroni informed Joseph that God had a work for him to do and told him that a record of the ancient inhabitants of the American continent was buried in a nearby hill. He stated that the record contained the fullness of the gospel of Jesus Christ. Joseph Smith later retrieves these gold plates and translates them into English by the inspiration of God. (remember this point for later)
David Whitmer, one of the three witnesses to the Book of Mormon, made this statement about how the book of Mormon was translated:
"I will now give you a description of the manner in which the Book of Mormon was translated. Joseph Smith would put the seer stone into a hat, and put his face in the hat, drawing it closely around his face to exclude the light; and in the darkness the spiritual light would shine. A piece of something resembling parchment would appear, and on that appeared the writing. One character at a time would appear, and under it was the interpretation in English." 
Oliver B. Huntington recorded in his journal that in 1881 Joseph F. Smith, who became the sixth President of the Mormon Church, taught that the Lord gave Joseph Smith the exact English wording and spelling that he should use in the Book of Mormon:
"Saturday Feb. 25, 1881, I …. Heard Joseph F. Smith describe the manner of translating the Book of Mormon by Joseph Smith the Prophet and Seer, which was as follows as near as I can recollect the substance of his description. Joseph did not render the writing on the gold plates into the English language in his own style of language as many people believe, but every word and every letter was given to him by the gift and power of God. So it is the work of God and not of Joseph Smith, and it was done in this way … . The Lord caused each word spelled as it is in the book to appear on the stones in short sentences or words, and when Joseph had uttered the sentence or word before him and the scribe had written it properly, that sentence would disappear and another appear. And if there was a word wrongly written or even a letter incorrect the writing on the stones would remain there. Then Joseph would require the scribe to spell the reading of the last spoken and thus find the mistake and when corrected the sentence would disappear as usual.” 
Just one more quote; this one from Joseph Smith himself:
“I told the brethren that the Book of Mormon was the most correct of any book on earth, and the keystone of our religion, and a man would get nearer to God by abiding by its precepts, than by any other book.”