I've tried using both. If I like the URL for the app (http://apps.facebok.com/myapp) it counts the likes, but I don't get content from the community page. The community page was generated from the developer app tool for this app. If I use the URL for my generated community page, I do get posts to the community page. My only issue with that is Facebook automatically offers to share the URL when you like it now and the URL that gets shared doesn't take users directly to my app. Is there a setting that I missed to make this all work with the APP URL?
After much research (and looking at source for other Facebook applications) the answer is to create a community page named the same as you application and in your application use the like button with the URL of the community page. If you set it up properly, the community page can have a link to your app if it is on Facebook.
It feels like a huge mistake on Facebook's part to not have a better connection between the page URL and the community page. Perhaps this will improve in the future.