No, the configuration.php file should be unwriteable. That's for security issues and is normal for Joomla.
As for the issue, can you check the following things:
* In the JFBConnect - FB Wall Integration plugin that the "Push Status Updates to Facebook" is enabled. That's the default, and would be surprised if not, but verify.
* In the plugin manager, try setting the JFBConnect - FB Wall Integration plugin to the highest position (lowest number) so that it executes earlier than the others. You can also try last, just in case, and even compare the ordering to the site that does work to see if there are any other differences there.
* In the JomSocial Configuration area, under Site, check your setting for "Activity Display Privacy". I believe either setting should work, but compare with your other site. Just so you know, any posts marked as 'private', 'friends only' or anything other than 'Public' when posted will *not* be pushed to Facebook. Our plugin will only push posts marked as public.
* Finally, in the JFBConnect Configuration -> Facebook API tab, try enabling "Show Facebook API Errors on Front-End". After you post the status, refresh the page and see if any errors appear... if so, let us know.
Hopefully, one of the above will help narrow things down.. definitely still unsure of what the problem is, but will do what we can to help fix it!
Alex