I was brought up with the Bible, but had to adjust to the concept of additional scripture in the Book of Mormon. An obvious difference I noticed was the form of writing: The King James scholars write differently from Joseph Smith. But this difference is just superficial. Looking for what is the same and what is different in the people and their doctrine is the important thing, along with the value of its teachings to us.
The BoM (Book of Mormon) people prior to Christ had a doctrine somewhat closer to the gospel in many ways than those in the OT (Old Testament). Yet it should also be remembered that although the OT was generally written toward a more spiritually backward people, it still contained the gospel principles for those who searched with the Spirit. The things Christ taught he could back up from the OT and other scriptures they had. The Pharisees knew this and couldn't argue it.
The part where the Bible comes to the fore is in the NT (New Testament) where it contains more books written after Christ, and by more authors. This gives greater understanding of gospel thinking in application. Additionally it has more of the teachings of Christ, details of the atonement in the garden, his death on the cross and the resurrection.
Yet the BoM has many invaluable lessons not stated so well or at all in the OT and NT. Not only that but the BoM demonstrates a greater average closeness to the Spirit consistently. In the NT Matthew has a great closeness to the Spirit. James and Peter also have this. And John feels an enormous love. Yet the OT is mostly very flat from this perspective. Isaiah shines forth and perhaps Jeremiah.
The BoM also has an advantage in not having so many of Christ's teachings, in that it makes his main message more obvious, thus not confusing the issue of what the fullness of the gospel actually is.
In spite of the lesser Spirit within it, I feel the value of the OT is generally ignored. God gave all the laws within the Law of Moses for a reason. Coming to understand those reasons brings us closer to understanding God and his way of working. It also helps us understand the gospel more completely, how to live it and how to advise others on principles of life.
Although Christ touches upon it in the NT, the BoM goes into greater depth about problems with lawyers and the legal system generally. It also goes into greater depth in regard political corruption. In that line there is also the demonstration of Mafia type problems.
Yet the NT gives a greater demonstration of the problems associated with Law of Moses type judgements. We see this with the Pharisees condemnation of Christ's actions. And his condemnation of their inability to judge people where their situations fall outside of the situations that God knew would USUALLY be the case, when he made the rules (eg. "But if ye had known what this meaneth, 'I would have mercy, and not sacrifice,' ye would not have condemned the guiltless" Matt 12:7).
Without the history of the OT, the NT and BoM would be hard to follow in parts. Mormon allowed for the existence of this historical information when he compiled the BoM (Eth 1:3-4).