Why is a Discount Rate Applied to Public Company Multiples When Valuing a Private Company?

The book answer is that liquidity is the reason, but I don't understand why liquidity would require applying a discount rate of 15% - 20%.

Also, why does a valuation for a public company give its implied share price? Doesn't the valuation still give the EV, and finding the implied share price is just another step that involves dividing by number of shares outstanding?

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