Why do the exported contacts differ from the contacts presented on the Advanced Search using Display Results as Related Contacts?
This is a known issue: https://issues.civicrm.org/jira/browse/CRM-15682

As a bug that affects relatively few people, the core team with their limited resources hasn't allocated someone to investigate and fix this.  If you're a developer or have access to one, this shouldn't be terribly difficult to develop a patch for; alternatively, if you don't have a developer, you may want to submit this to the paid issue queue.