I think, correct me if i'm wrong, that the differences are :
-how you open it (swf need a flashplayer and air doesn't)
-the saves locations
Well yes, but actually no.
AIR is itself essentially a flash player that can only run flashes made using ActionScript 3. That means most made after around 2008.
AIR apps are usually packaged so that the SWF is considerably "baked in" to the runtime, but if you look in the AIR app's install location you'll see there's still a special SWF in there that won't run properly in a normal flash player. That's because of the file handling differences, which are accounted for when the game is compiled.
It is meant to run SWF's as if they were stand-alone applications, so it has more freedom to access system files than a normal flash player SWF does. It has special extra 'methods' (think subroutines or something) that let it handle files more flexibly, but the regular flash player knows nothing about those routines and will panic when it can't find them.
In the end they both should work (if you use a Windows/Mac OS system, at least). I use SWF's myself in the stand-alone Flash projector adobe publishes, but I did make my own AIR builds for mobile devices. if fen & co. ever decide to drop source code again (still waiting on the sources for the public build from 8 days ago), I'll be making more.