You have to delete/uninstall the application, remove the permissions and try again.
I wish I could tell you the exact problem with the application but I have not figured it out. Facebook and Twitter do not share information well so there are bugs that have not been fixed over the past few months.
I'm having the exact same problem - goes through the approval process fine, but nothing appears on the page. As you say, profile is working fine, just pages. I tried different accounts and browsers, as well as a few tricks, same result. I can only assume it's a bug as the whole Twitter Facebook thing has been shifty this month. One workaround is to use the facebook app Selective Tweets. This will post to pages; however unlike the native twitter app, each tweet you want to go to the facebook page you must end with #fb, hence the whole "Selective" thing. But at least they show up (and the #fb tag won't show on the page).
Best Answer
Depends on which way.
Facebook
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Twitter is possible via http://www.facebook.com/twitterThe opposite way there is no official channel but many apps are available such as Involver.