In these days of precise economic modelling and sometimes wafer thin profit margins airlines are more able and very keen to get the "best" aircraft for their particular situation. If one of the 737 or A320 were truly "best" then you would see the sales skewed much more to one or the other. The fact that they both sell well shows that neither is better when everything is taken into account and the answer is "It depends...." Aquisition cost and leasing v finance deals available, depreciation and eventual disposability, availability and waiting time for slots, expected future model additions/enhancements, current and future fleet commonality, engine availability and commonality, (re)training costs, length of routes and frequency of segments flown (fuel burn v maintenance), type of cabin configuration required, range of cabin sizes required, cargo requirements, baggage loading issues (loose v container), performance at marginal airfields (hot and high, short field, ramp space restricted), maintenance costs, local maintenance expertise, previous product (dis)satisfaction, political issues (protectionism, embargos, 'special' deals for regional penetration etc.). Those are just of few of issues and facts airlines must consider. There are probably 100's more.
The Boeing 737 is still superior in popularity, because it was introduced so long ago, and all the long lasting airlines most likely have gotten their hands on a 737 at some time. IF the 737s in your fleet are getting old, well, BAM! the 737NG and 737X is to the rescue. The thing with the 737 is, is that it just keeps on coming, and most airlines that have operated 737s for a decent amount of time (which is most large airlines) would probably prefer to stick with the 737 with their future models.
Sorry I don't agree with that at all. I'm not taking side as I think they are both very good but... there are MANY large airlines who have switched from quite large fleets of 737 to A320 eg. Lufthansa, British Airways, Air France (old)United, US airways, Air Canada, Easyjet just to name the first few that come to mind . Many also have both as the economies of scale kick in with very large fleets and mergers add to their fleet complexity (American, Delta, the new United). I think my argument above is proven by the fact that all three of those large American carriers I mention have BOTH in their fleet, will continue to operate BOTH for a VERY long time and are actually adding, or about to start adding new examples of BOTH. Off the top of my head only Southwest, Ryanair , KLM, Alaska and Qantas come to mind as large airlines who have stuck stubbornly with 737s but I'm sure there are many more too. I am struggling to think of any who have been predominantly A320 and have then moved to 737 - Only Air Canada come to mind with their recent order for 737Max.