Why does a public company issue a prospectus but a private company does not do so?

Because the Securities and Exchange Commission requires them to do so (or does not require them to do so, for private companies).

Basically, what it boils down to is that if you want to participate in the Stock Market, you have to agree to a whole bunch of rules as to what you're required and not allowed to do. If you don't care about participating in the stock market (because you're privately held), then you you're not subject to the same rules. A private company could in theory publish a prospectus, but it would mainly be a waste of money; the people who own the company are most likely intimately involved in it and already have a very good idea of how the company is doing, who the other owners are, etc.