I've had the A320 a couple months now (although I haven't had a lot of time to play FS recently and dive into all the technical details) and bought the 737 last night, so here are my first impressions:
The 737 seems to have a "warmer", friendlier feel to it, the manuals seem easier and more interesting to read, and the FMC even has its own help section you can click on. And you can even load up the FMC as a stand-alone application while you're not in FS, to practice on it, make up new flight plans, etc. The A320 seems "colder", less welcoming, the manuals seem more "dry" and technical to read.
The A320 actually includes the A319, A320, and A321 models. However, you have to pay extra to get more liveries off their website, and you even have to pay more than once for the same airline if you want it in 2 or 3 of the models instead of just one. With the 737 you can download any number of liveries for free, plus they provide a template to help you make your own, which many people have enjoyed doing.
The A320 has a state-of-the-art 3-D virtual cockpit you can pan and move around, while the 737 (being designed for FS2000 originally) uses photographs of the cockpit at set angles. However, with the 737 you also get cabin and wing views. (again, from photographs)
Finally, something I think is really cool, the 737 lets you customize the seating layout, moving groups of 2 first-class seats or 3 coach seats around wherever you want, you can even turn them backwards if you want club seating. You could have no seats at all, or all first class, anything in between, however you want. After you decide where the seats go, you can click on seats to put people there, or let the computer fill them randomly, and then the weight of the plane is set based on all this. Too bad you can't put in beds, couches, desks, etc. like on John Travolta's private 707. Maybe Dreamfleet should make a BBJ next. :-]
So basically, I like both of them a lot, they each have their own little advantages and disadvantages. If you can only afford one, then I'd say get the A320 if you like to spend most of your time in the cockpit doing everything seriously and by the book, or the 737 if you like watching your plane from the outside and/or think it would be fun to repaint it yourself.