I'm type-rated on both the 737 and A320. I flew the 737-300/400 and 737-800 and A319/320/321.
So I can compare all these aircraft. I'm sick of all those 'pilots' who never saw an A320 from close or flew one and giving their opinion about the flying qualities of this beautiful machine.
My choice : Airbus A320.
Both the 737 and A320 are nice to fly but take a closer look to the cockpit and you know why I prefer the A320. This a a nice ergonomic cockpit where you have place to put your pilot-case without ruin it, where you can eat your meal without that stupid and big control wheel in just front of you (or in your meal while the aircraft is turning).
Fly-by-Wire ? Never had the feeling that I was not flying the airplane. The Concorde (analog) has it, the 777 has it, the Tu-204 has it and most military aircraft today has it.
The A320 has a cockpit where you can talk to eachother without yelling. Ever flew a 737-800 at M.79 ? Never heard so much wind-noice in a cockpit. Lot's of pilots at our company flew with ANR headsets !!
The 737 is designed in the 60's and you can see it !!