On my mission, my companion and I had a grand idea—set up a table in the major courtyard of the college campus where we served. Many people, adversarial or curious, would stop by and check out our booth.
One girl asked me why we needed the Book of Mormon when we have the Bible already. My answer at the time was woefully inadequate, and we both knew it. Today I saw well-meaning people wandering around a park, trying to spark up religious discussion while offering free water. I did not speak with them (contention is of the devil, and I became mildly aggravated just thinking about talking to them). But the encounter reminded me of my earlier experience on my mission. Below is my current answer to the question, "Why do we need the Book of Mormon?"
Living Messengers
The Bible was written thousands of years ago by prophets, apostles, and saints who are currently dead (or resurrected; unavailable either way). While their words are true, they did not live long enough to pass on their authority to a new generation of prophets. The Book of Mormon comes to the world with ordained ministers attached. Not only did God provide new information, new scripture; He also provides living, ordained prophets, apostles, and missionaries with the authority to baptize and lay hands on for the receipt of the Holy Ghost. This has not been true of the Bible since the time of the Apostles. To accept the Book of Mormon as true is to accept the principle of modern revelation.
Indivisible
Another useful function of the Book of Mormon is its indivisibility. By that I mean that it cannot be accepted in slices. The Bible can be approached as the word of God, a history of true miracles and divine theophanies, or it can be accepted as a partially true narrative with fantastical lies mingled with verifiable historical facts. You can learn about culture and custom, say the critics, but there are no visible footprints on the Sea of Galilee. So the Bible becomes a buffet, where people take what they want, and leave what they don't. Not so with the Book of Mormon. It is either completely true, or completely false, with no convenient middle ground. Also, it cannot be misinterpreted or used to start new religions, since it comes with living apostles. Many splinter groups have left the Church since Joseph Smith died; none of them I know of use the Book of Mormon in any significant way. The Bible is viewed by many as a do-it-yourself build-a-religion kit; the Book of Mormon lacks that option.
Geographical Gathering
Because the Book of Mormon comes with living messengers, authorized to baptize and build the Church of Christ again on the earth, it becomes something else the Bible alone cannot be: an invitation to literally, physically gather. The Bible may stir longing to gather, but where? The Old Testament gives us the law, and demonstrates obedience, and consequences of choices. The New Testament spreads the message that Jesus sacrificed for us, and fulfilled that old law. But the tens of thousands of divergent denominations, fragments of Christianity scattered the world over, are clear evidence that the "one Lord, one faith, one baptism" Paul described are not widespread. The Bible has been on the earth for nearly two millennia now, and the problem of division and disagreement among sects is still rampant. Isaiah spoke of "a remnant" of Israel returning. When will that happen? And how? The Bible is apparently not the answer to the problem of gathering Israel; the Book of Mormon is.
The Book of Mormon is like a dinner bell rung at the end of a long, hard day, inviting weary children home to sit at the table of the Lord. Not only does the Book of Mormon come equipped with genuine messengers from God—it also comes with geographical promised lands, wards, stakes, and Temples where saints can gather literally and physically. It not only brings home the Lord's "sheep," who are familiar with His voice; it also feeds them til the Savior comes.
The Bible sends out the call, but it lacks a specific address for gathering, and the confused response to its message manifests itself as fractious sects, all Christian, yet all separate.
Personal Revelation
Finally, the Book of Mormon comes with the promise of personal communication from God, even an invitation to obtain a witness through the Holy Ghost. A religion that does not involve personal communication with God is a farce (early Church presidents expressed this idea). Why should anyone join this Church rather than others? Ask God. If you cannot get any response, you probably should not be going to any church at all.
The Bible describes people interacting with God; only the lost invest themselves in arguing over what God said anciently. The Pharisees knew the scriptures; Jesus knew God. The ancients set the example for us, but how few have actually followed it! Scripture study is important, but it is useful only as it leads to revelation, personal interaction with our Heavenly Father.
Those who approach the Lord with an I-will-do-anything-you-ask attitude are the ones who get revelation more readily, I believe. No pope, no saints, no middlemen of any kind—secure knowledge for yourself, on your knees. That is the invitation of Joseph Smith and his successors; it is the invitation of the Book of Mormon. Anciently, religion and politics and culture were the same thing; you were born into a civilization, and adopted the customs of your ancestors and community. Religious adherence was compulsory. Not anymore. Now we have a buffet of philosophies, religions, sects, gurus, charlatans, snake oil peddlers, etc., to choose from. We have the luxury and the time to get distracted. That is the world we see today. Rich people trying on religions and philosophies like sweaters, all window shopping.
The Book of Mormon presents the obvious, yet daunting solution: Ask God. My witness is that God answers our prayers, if we pray in faith and humility. Willingness to comply with revelation is the main key, in my opinion, to qualify to receive it. It is also my witness that the Lord has confirmed to me, through personal revelation, that the Bible, Book of Mormon, and other restored scriptures, are true. Joseph Smith was called of God. There is a living prophet on the earth today, and the authority to administer the ancient ordinances is restored again. I feel it a privilege to sit quietly in the morning and pore over the Book of Mormon and Doctrine and Covenants. I hear my Savior's voice in the words, and I am grateful for all the scriptures He has given us.
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15 November 2015, Update: There is an excellent new site about the Book of Mormon called bookofmormoncentral.org. The general concept is to create a comprehensive website about all Book of Mormon scholorship, everything ever written (or sung, painted, sculpted, etc.) about it. There are scores of excellent articles about linguistics, archaeology, and principles.