National corps are the same as State except they cannot be bid on by a CEO if you allow outside investment in your country (on the settings page).
Once a corp goes public you cannot transfer cash to it, so the only corps that should go public are the ones that have a constant profit.
When you IPO your country will receive the cash for the shares you sell.
As a State corp your country receives 100% of the profit form a corp. As a public corp you still receive the taxes but only the percentage of profits based on your % of ownership. Also public corps cannot be bid on by CEO's and are immune to earthquake, war and civil unrest.
So decide if you want fast cash or long term profit.