Book value per share, strictly, is total shareholders' equity under GAAP, minus anything owed to preferred shareholders, divided by common shares outstanding. That's the one in the balance sheet and it's the one every filing has to report the same way.
Economic book value is a company-defined figure. It's not a standard measure, and different managers compute it differently. The usual reason it exists: under GAAP some assets are carried at fair value while some liabilities or servicing assets are carried at amortized cost, so a manager will argue GAAP understates what the portfolio is really worth and publish a version where everything is marked to fair value. The adjustment is often about mortgage servicing rights or about securitized debt held at cost. Sometimes the argument is fair. It is always the company's own arithmetic, so read the reconciliation table that has to accompany it.
When the two disagree, use GAAP book value for comparison across companies, since it's the only one computed on a common basis, and read the economic version to understand what management thinks the market is missing.
The reason management dwells on it: dividends can be paid out of a lot of things, and book value is the running record of whether the business is preserving your capital while it pays you. A company that paid out 12% a year for five years while book value halved gave you your own money back with extra steps. So track the sequence of book value per share alongside the dividends paid over the same period, and add them together. That sum is the honest scoreboard.