It shouldn't be logging you out of Facebook when you close your browser. That's never been a capability of the software.
We're updating the login routines quite extensively for the new version (due out within a day or so). This may help fix the problem.. However, try recreating this issue on our demo site. It's using 2.1.2 and if we log into Facebook with the "Keep me logged in" box set, go to the demo site and log in there, then close the browser... when we start it back up if we go to Facebook, we're still logged in.
Let us know if you're seeing something different on the demo site, because that is strange.
The next version does handle log-outs better (you can even use any login modules 'logout' button without forcing you out of Facebook!)