I know Travolta is some official spokesperson or ambassador for Qantas but I find it odd. Its I guess why he is allowed to retain the livery on his 707. Apart from starring with Olivia in Grease I don’t get the connection with Australia for him. Its also odd to see Qantas livery in Florida since I don’t think they fly there but I could be wrong. It sort of reminds me of that funny scene in Rain Man.
I think the Braniff livery on 707-138 was cooler than the orginal QANTAS livery. Braniff bought 707-138's second hand from QANTAS in the mid 1960's. Braniff always tended to get high performance hot rod type aircraft like the 720-27, 707-227, and 707-138 due to their South American and Central American routes. Mexico City has an altitude of over 7,000 feet, and La Paz, Bolivia has an altitude of over 13,000 feet. After Braniff bought PanAgra, they inherited PanAgra's DC-8-62 orders from Douglas. These planes had the same engines as the DC-8-71 and DC-8-73 but had a shorter fuselage that weighed less. This allowed for the longest ranged passenger jet in the world prior to the 747, but also allowed higher performance take offs from high altitude airports with less weight.
http://www.airliners.net/open.file?id=0221086&size=M
Unless Qantas flies into Miami or Orlando, that may be true; however, I believe Qantas does fly into LAX - Travolta's main home.
I think he and Qantas have this odd connection, because he wanted to buy a larger jet and they happened to have one for sale. I think it's as simple as that.
Since Travolta is recognized world-wide, Qantas may have seen the advantage in using his image to garner more sales in the US - to compete with United, Air New Zealand, Singapore Airlines, and American, etc - on direct flights from LAX to Sydney.