The recent merger between the two has now produced a hybrid which will alternately be known as the "Bobus," or the "Airbing." The Bobus will continue to be known by it's cute little bottle nose and the crews that swagger around with chart cases covered in "if it ain't boeing, I ain't going," stickers. The Airbing is undergoing a redesign, and will soon see self launching using larger rubber bands and a crew of six; three to pull it back and hang on, and three to wind one up internally for the long flight home.
The easiest way to tell is to check out the header on the passenger safety briefing cards when you buckle in, right after you turn off your cell phone and take a couple of cough drops so I don't have to listen to you for the next four hours. If it says "Airbus," it probably is...but could just be a misprint. If it says "Boeing," you can bank on it.
If you see one taxiing by and the pilots are wearing Berets, chances are it's not a Boeing. Changes are you want to avoid shaking hands with them after the flight, too. Or spending too much time with them at the bar. If you see one taxiing by and the crew are wearing crush caps, haven't shaved in a few days, look haggard and one crewmember is possibly dead, it's a boeing, but 60 years out of date and on the wrong side of the pond. Check your time-space continuem for proper function.
If you say "airbus" too loudly and someone smacks you upside the head with a wrench, you're probably in the Boeing production plant, and had best scoot for an exit forthwith before someone else does something more than say "hello."
If you happen to be in France when you say "Boeing," you'll probably see more white flags tossed at your feet than an epileptic in Pennys during a white sale. Not that the airbus doesn't sell well, mind you...it's the mindset of surrendering to anything, any time...and it's not their fault. Several millenia of defeat will do that to anybody.
If the aircraft is old, it's probably a Boeing. If it's not it's probably an airbus.
If you get a chance, ask the pilot. If the pilot replies "Boeing!," changes are that he's correct. If he replies "Boeing?," chances are that it's an airbus.