In the API result (http://developers.facebook.com/docs/reference/fql/link_stat/) there are two counts about comments:
comment_count: The number of comments users have made on the shared story
commentsbox_count: The number of comments from a comments box on this URL. This only includes top level comments, not replies
What is the difference? Does the comment_count include the commentsbox_count?
They are quite different from each other.
comment_count refers to the number of comments made inside Facebook on a shared story, where as commentsbox_count refers to the comments made on the external site via the comments plugin.
Sites can use the comments plugin to easily put up a comment box on their website without having to implement it themselves. The commentsbox_count can be queried to get the # of comments left on each URL there.