The book value of an asset equals the price that you paid minus any depreciation in value of the asset. Book value either stays the same or falls.
The current price on the open market rises and falls depending on several factors that have nothing to do with the book value of your asset. It can increase or decrease after you buy the asset.
Fair value is used to figure replacement cost. Book value isn't used when replacing assets or figuring the amount of insurance needed on your current assets, as replacing an asset involves buying it at market price. Fair value indicates whether your asset is priced too high or too low.