I don't think there is any difference as far as "names" go for companies. You know, Delta airlines, British airways, Same thing. It's what the companies choose to call themselves. The only difference is the meaning of the two words, however, when used in this context, they have the same meaning.
Airlines is usually referred to the companies in the aero-transportation business. For example: Delta Airlines, American Airlines.
Airways are the routes that planes use to move from place to place.
But I have seen some companies suing airways for their names like: Tiger Airways, Kuwait airways, and some using airlines like Garuda Airlines etc.
I see, well then, I really don't know. I don't think there is much difference, but the only thing that I can think of is copyright issues.
Airlines is for like airplanes and airways is for like radios and tv's