I am going to compare Boeing 737 and Airbus a320 as an amateur aviator.
I simply see Boeing and Airbus like a DSLR camera, one of them in on Manuel mode (Boeing) and the other one is on Auto mode ( Airbus ).
First of all lets see how we can differentiate this two beautiful bird.
On top there is A320 and on bottom 737-800.
Firstly, as you can see B737 has more pointed nose (nose cone) compare to A320.
When we give a closer look to their landing gear, it can be easily said that a320 is slightly higher then 737.
On the wings the difference is more obvious, B737 has winglets like fins on the other hand A320's winglets like a fish's tail.
There is also a difference on engines, remember A320 is higher then a 737, thats why 737 engine covers are flattened due to ground clearance. A320 on the other hand have round shaped engine cover.
Of course those statements are valid for Airbus A320 and Boeing 737-800NG. New Airbus A320 Neo also has winglets like fins ( They are called sharklets ). However, there will be still difference because the new Boeing 737 Max models will have an unique winglet design.