The Boeing Company is an American corporation that specializes on manufacturing of aircrafts, space and military technologies. There are many large and well-known manufacturers of aircrafts in the world like Airbus, Bombardier, Embraer, Saab AB, ATR. And Boeing is one of them. Boeing 737, Boeing 747, Boeing 757, Boeing 767, Boeing 777, Boeing 787 and their modifications are the most recognizable aircrafts in the world.
Airline is organization that provides services of transportation of passengers and freight by air transport. Airline rents or has its own aircrafts for providing services of transportation. It can use aircrafts from different manufacturers: Airbus, Boeing and others. Some of well-known airlines are: Southwest Airlines, United Airlines, American Airlines, Delta Airlines.
The Boeing Company is manufacturer, and airline is provider of services. Boeing just produces aircrafts, and any airline can buy aircrafts from Boeing and use them for providing services of transportation. Boeing does not provide services of transportation. This means that expression “Boeing Airlines” is incorrect. Boeing Airlines at the moment does not exist. May be in future Boeing will create its own airline.