Why do Boeing and Airbus have distinctively different nose designs?
The answer to this is, in part to do with corporate culture and part aerodynamics.

The corporate culture and history part is that Boeing have always built their noses that way and senior engineers have a tendency to return to designs they have used successfully before. If you look at the nose of a 747 and the nose of a B-17 you'll see some distinctive similarities that follow through all the major Boeing aircraft. This isn't surprising as lead designer of the 747, Joe Sutter, started working for Boeing on the 707 project under the direction of the engineers that designed the early flying forts.
Airbus is a newer company with a different corporate culture. They tend to embrace new design techniques like CFD computer modelling to a greater extent and this leads to slightly more efficient but less aesthetically pleasing nose and wing designs.   

The science side is related to the speed that modern airliners travel at (around 0.85mach). At these speeds aircraft begin to encounter a phenomenon called 'Wave Drag' which is the incremental build-up of compressive shockwaves along the wings and fuselage as the aircraft approaches the speed of sound. The effect of wave drag is to drastically increase the overall drag affecting the aircraft, increasing the power output required from the engines.

There are several methods of combating wave drag, the first of these is the swept wing that has been a feature of trans and supersonic aircraft since WW2. The optimum angle of wing sweep is determined by the cruising speed of the aircraft. A more recent innovation is the Transonic or 'Whitcomb' Area Rule which states that: 

"Two airplanes with the same longitudinal cross-sectional area distribution have the same wave drag, independent of how the area is distributed laterally"

Wave Drag can be reduced by attempting to match the cross sectional proportions of an aircraft as closely as possible to those of a Sears-Haack body (an aerodynamically perfect shape for supersonic flight) This is why Airbus planes have blunter nose. it is also why there are strange bulges under the wing roots of the A380. they are both attempts to make the planes cross-section conform to a Sears-Haack body more closely.