This is interesting, though I would have been more interested in fare differences when booked across these different regions (maybe worth looking into!).
My suspicion is that it's because Australia has quite lax advertising laws. In fact, the ACCC specifically lists "puffery" as an exception to false advertising, See information
here. This basically means that you're allowed to say (as per the ACCC example) that you have the "best steaks on Earth" without it being considered misleading.
This is pretty relevant to the Qantas ad, because statements like "Australia's favourite airline" are clearly puffery, "Australia's most popular airline" may also fall into this category. Even though I don't doubt that Qantas has the highest passenger numbers out of any airline operating in AU, "most popular" is generally not well defined.
I suspect that the US has stricter advertising laws where "puffery" isn't as well accepted. Thus, they go with a much more subdued claim "one of Australia's most popular airlines".