Offshore companies:
-Cannot have physical presence in the UAE. Their office must be outside the UAE.
-Are not authorised to issue resident visas.
-Cannot directly operate any business activity in the UAE. However, they can be a shareholder of any UAE mainland or freezone company, enabling them to enter the UAE market and conduct business activities through such subsidiary companies.
-Often used as holding or asset companies for other business organizations which operate inside or outside the UAE.
-Not subject to VAT registration
Freezone companies:
-Can do business in the freezone, but not in the mainland.
-Can issue resident visas for shareholders and employees
-Typically required to have an office in their freezone.
-Subject to VAT registration (if they meet the criteria)
And well, offshore companies have been quite unsual lately, and they are - as far as I know - only issued by three UAE jursidictions: RAK International Corporate Centre (RAK ICC), Jebel Ali Free Zone and Ajman Free Zone. It is possible that "everybody" is aware of the freezone rules, since freezone company has been the totally dominant company form (along with mainland company), but that few people know about offshore companies and the rules that apply to them.