Public ownership delivers a certain level of prestige that private companies do not have. Going public provides owners and founders liquidity for their holdings in the business. For entrepreneurs approaching retirement, an IPO makes estate planning dramatically easier and a lot more enjoyable. Public companies enjoy a higher valuation than private enterprises. That's less than half the public valuation, largely because there is little liquidity for investors in a private company.
Public companies are more attractive acquisition candidates.
For some companies, the strategy is to grow to the point where they become an attractive middle-market acquisition candidate and then to sell out. Going public usually makes you a millionaire. Even if you initially don't get the price you want, playing the waiting game can be profitable. Another reason that so few deals close is that most investment bankers who underwrite IPOs are unwilling to take risks on perfect strangers. There is plenty of legal exposure even when they've identified all the warts.