It's not that AT&T let's you keep the broken iPhone. If you file a claim you go through the third party insurance, which in this case happens to be Asurion. And no, if you are getting the phone swapped out through the insurance you have to send in the non working phone within a certain period of time, otherwise they bill you the full price for the replacement phone. Since AppleCare+ doesn't cover lost/stolen, I'd say it is a matter of why you need a replacement as to whether it is cheaper or not. If you lost or had your 16GB iPhone 4S stolen with AppleCare+ (example), you could end up paying out $599. + tax for a brand new one. With Asurion, you'll get a replacement for $199. Btw, yes, AT&T's wireless insurance also covers warranty issues as well, so people that still have this on their line after the first year can have their device replaced if their is a warranty issue with the phone.
AT&T's wireless insurance as of Oct. 7th for the iPhone is $6.99/month, which comes out to $83.88/year. The deductible is $199. per incident with a max of 2 per year. Having a damaged phone replaced with AppleCare+ is definitely going to be cheaper in the long run, however, needing a replacement due to lost/stolen is cheaper with Asurion.