Why do Airbus aircraft typically have less wing flex than Boeing aircraft?
It's not a super scientific statement. There may be some half-truths referring to half-truths inferring that some design is better, just for show business sake. 

"The Boeing wing is way more aerodynamic than the Airbus wing" is not something that has any value at all. All wings are by definition aerodynamic, there is no sliding scale. All wings are the result of a design process, balancing many input parameters. If an aircraft manufacturer cannot get the aerodynamics of a wing right, they have no place in the aerospace industry. Both Airbus and Boeing make very fuel efficient and safe aircraft that compete head to head in the market place.

Wing flex is one of these things that the structural engineers must cater for, after the aerodynamicists have had their go at the external shape: the planform, aerofoils etc. It can be considered a secondary design parameter that:


provides dihedral in flight, for stability when banking;
acts as a spring against turbulence;
can cause grief with flutter;
may provide difficulty for the ailerons to operate.


For a given wing shape, wing flex is the result of the ratio of strength and elasticity of the wing material. Composites are the latest in aircraft material, being lighter and stronger than aluminium. It is five times stronger than aluminium, and flexes about half when under load. But because it is stronger a lot less material can be used to make the wing lighter - even though a certain amount of carbon fibre flexes less than the same amount of aluminium, there is simply less of it. The ratio of strength over flex of carbon fibre is much higher than that of aluminium.

All aircraft with wings made of carbon fibre flex a lot more than aircraft with wings made out of aluminium. Both the A320 and the B737 have aluminium wings. For a given material and wing area, wing flex is a function of 


Aspect ratio: a long, slender wing flexes more than a short, stubby wing. Or the long aluminium wing of a large airliner flexes more than the shorter wing with the same aspect ratio of a smaller airliner.
Wing thickness. The thicker the wing profile, the less sweep angle needs to be used to delay drag divergence. This is actually a sign of an aerodynamically more advanced wing, and the thicker profile makes the wing bend less


Those sort of comparisons make much more sense than "the wing of a Boeing is much more aerodynamic than that of an Airbus". That's only showbiz.