Boeing aircraft are controlled by pilots. Airbus aircraft are controlled by engineers sitting at their desks in France. If something goes wrong, the pilot dies, so he is strongly motivated to do whatever is necessary to survive. If something goes wrong, engineers sitting at desks are strongly motivated to cover their backsides and claim that it's all the pilot's fault.
Therefore I trust Boeing more than Airbus.
There are other reasons as well, having to do with subsystems that don't fit because the engineers never coordinate their efforts, wings that snap off at less than their design maximum loads, questionable ways of obtaining contracts, choosing manufacturing facilities based on politics rather than technical desirability, and the like, but suffice it to say that, if I have a choice, I'll always prefer a Boeing aircraft.
Yes, both companies build extremely safe airplanes, but if I have to choose … if it's not Boeing, I'm not going.