INFO: Mozilla has discontinued FTP service as of 2015-08-05, see the following blog post for details:
Polls Mozilla's FTP Server for latest "Firefox for Android" release, downloads it and -- after comparing the versionCode -- installs it (if needed).
Currently the app is an activity with a two-button GUI. Checks are made when the "I am feeling lucky"-button is tapped. The other downloads the last version known on release date. However, the plan is to move this actually to an Android service to perform periodical update checks.
However, Mozilla plans to shutdown the FTP server, so we cannot rely on this update method. I opened a ticket about non-playstore updates with Mozilla:
https://bugzilla.mozilla.org/show_bug.cgi?id=1192279
Another related issue is tracker at:
https://bugzilla.mozilla.org/show_bug.cgi?id=1220773
FFUpdater -- a simple Android app to update Firefox. Copyright (C) 2015 Boris Kraut krt@nurfuerspam.de
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.