The compatibility mode is related to APPSEC-1034, addressing bypassing custom admin URL
From the SUPEE-6788 patch detail page:
Admin Path Disclosure - APPSEC-1034
CVSSv3 Severity: 5.3 (Medium)
Attacker can force showing admin panel login page regardless of admin
panel URL by calling a module directly. It makes it easier to try
automated password attacks and exposes admin URL on the page.
The best way is to update the extensions and disable compatibility mode:
1.) All reliable extension vendors updated their extensions so you just have to look for the latest releases.
2.) Update your own extensions with the information from the technical details page
See also here: