Confusion over the Lamb’s Book of Life and the Book of Life
The question concern the differences between two books that have very similar names: the Book of Life and the Lamb’s Book of Life. It is because they have similar names, along with doctrinal error about election, these two books are confused.
When undertaking a study of any type one must be vigilant to collect information about what scientists call confounding factors. These are matters, or things, or actions or results that might obscure or change the observation we are making. When reading about these two books there are two confounding factors or elements that must be considered:
Eternal security – The Bible shows absolutely, in particular in the Letter to the Romans (Romans) that any person believing on the Lord Jesus Christ is saved, and that salvation is everlasting. Put another way, and in terms of the Saviour – Jesus Christ, the sacrifice he made on the cross has paid our debt in full for past, current and future sin. Eternal security has bearing on the question because someone saved, and thus having eternal security, cannot have his or her name blotted out or removed from any list, in particular one that lists the saved.
Election – concerns the action of God for a person in relation to their salvation before the creation of this world. To an unbeliever election is totally obscured and is meaningless. For the believer, in their maturity, realisation of their status of being elected will humble them before God. Election has a bearing on the question because an elected person cannot be removed from any list, which lists those saved. Election of those that will be saved occurred before the world was created – this status cannot be changed. 
The Book of Life
The Book of Life contains the names of every person who was or will be born according to the Psalmist, David.
Your eyes saw my substance, being yet unformed.
And in Your book they all were written,
Note the small word “all”. The context is plain – it means everyone. Thus this book cannot be a book of elected persons, because, since every soul is recorded in the book and the Bible states some will not be saved, it would contract the Bible’s definition of election (which only pertains to saved persons). That a name can be blotted, or removed, from the Book of Life, we find to be true, evidenced by Revelation 3:5.
He who overcomes shall be clothed in white garments, and I will not blot out his name from the Book of Life; but I will confess his name before My Father and before His angels. (Revelation 3:5)
The verse states God will not blot out a certain class of persons from the Book of Life. This means there must be at least one other class of persons that will be blotted out. Note the affirmation: ‘not to have’ one’s name removed, thus logic dictates that names must be able to be removed from the Book of Life. The class that will not have their names removed are those that are saved, and the class blotted out are the unbelievers. And again Scripture confirms this – Psalm 69:28 speaks, not about those that are saved, but about the wicked and those of iniquity who will have their names blotted out:
Let them [those full of iniquity] be blotted out of the book of the living,
And not be written with the righteous. (Psalms 69:28)
The Lamb’s Book of Life
Another book is also mentioned in Revelation, and only in Revelation; and it is clear why this is so. This is because this book contains the names of every individual that is, and will be, born again, and in particular in the context of those that will not worship the Antichrist (whose actions are spelt out in the Book of Revelation).
All who dwell on the earth will worship him [the Antichrist], whose names have not been written in the Book of Life of the Lamb Slain from the foundation of the world. (Revelation 13:8)
In this book we have the names of every person who has been created, and thus given a soul, and who is saved (born again). It is this class of person who will not worship the Beast. In other terms, these are the elect of God, because these names were written before the foundation of the world and are saved. Revelation 17:8 tells us these will not worship the Antichrist, i.e. beast.
The beast that you saw was, and is not, and will ascend out of the bottomless pit and go to perdition. And those who dwell on the earth will marvel, whose names are not written in the Book of Life [of the Lamb who was slain] from the foundation of the world, when they see the beast that was, and is not, and yet is. (Revelation 17:8)
Note that again, like chapter 13, the writer indicates the names were written in the book from the foundation of the world, emphasising these are the elect of God.
Other names for the Lamb’s Book of Life
We have learnt that The Lamb’s Book of Life is not the Book of Life. We also learn that the Psalmist in Psalm 69 calls The Lamb’s Book of Life the Book of the Righteous. This makes sense, because spiritually anyone not saved (and thus has not been elected) is dead; dead in their trespasses (Ephesians 2:1, 5; Col 2:13) – indeed dead men and women walking because they will be cast into Hell (Revelation 20:15) – BUT those that are saved are made righteous by the blood of Jesus Christ (Romans 3:21, 25) – and their names are in a list of the righteous (and are therefore the elect of God (1 Peter 1:2)).
Let them be blotted out of the book of the living, And not be written with the [book of the] righteous. (Psalms 69:28)
The Lamb’s Book of Life, is the book of the elect and contains only the names of those who are born again. And because salvation is eternally sure (the atoning sacrifice of Jesus Christ is sufficient for all), no-one can be blotted out of this book. Furthermore, the only reason they can be elected is because of the atoning work of Christ who offered a perfect sacrifice – for this reason the book is referred to as the Lamb’s Book of Life (see also 1 Peter 1:19) – more correctly the Slain Lamb’s Book of Life. On the other hand the Book of Life is the list of everyone created and thus given a soul – in essence a list of all souls>. Many (most) will have their name blotted out, because they have not put their trust in the Lord Jesus Christ believed in him (John 3:16; Romans 3:21-31 etc.) - they will be cast into the lake of fire, called Hell.