Registered public stock is available to be traded by the public. It is registered with the states and the federal government in order to be in compliance with the law.
Unregistered private stock is stock that has not been registered. It is not available for public trading. Also, it's privately held which means it is not available in the marketplace.
The main differences are registered public stock has a daily price, a liquid market, and many investors. On the other hand, unregistered private stock has no daily available price, is not publicly traded, and is owned by very few shareholders.