Subjective selection of those changes to Firefox, which are probably relevant for admins or helpdesk.
How to read this:
When will this page be updated:

Sources: Most info was extracted from the release notes (also the ones for beta and aurora). If you want more, read the notes for developers, Compatibility Notes, Tracking, Meeting Notes. To be completely overwhelmed follow the links "complete list of changes" in each of the release notes.
click here for preliminary info for version 46 to 52
VERY PRELIMINARY INFO FOR VERSIONS 34 TO 38:
hotfix restore safebrowsing
43.0.4 undo blocking of sha-1 certificates, because many security devices and web interfaces of home routers stopped working
alternative: set security.pki.sha1_enforcement_level to 0
hotfix re-enable high-res youtube videos
43.0.3 undo blocking of NVidia DLLs, because it made the problem worse instead of fixing it
43.0.2 add sha-256 signing certificate, to meet new signing requirement
security fix
43.0.1 prepare to use SHA-256 signing certificate for Windows builds, to meet new signing requirement
43.0.0 many security fixes
turns off safebrowsing (accidentally)
will reject sha1-certificates starting January 1st 2016
42 ??
41 ??
40.0
Compatibility: faster Plugin-Initialization
apparently none of the beta-testers plays Farmville.
39.0.3 ??
39.0 ??
38.2.0 ??
38.1.1 ??
38.1.0 ??
38.0.5
Privacy: Keep track of articles and videos with ???Pocket???
Privacy: Share the active tab or window in a Hello conversation
Noteworthy improvement: Clean formatting for articles and blog posts with Reader View
38.0.1
a few fixes
see also https://mike.kaply.com/2015/05/05/firefox-esr-38-overview/
38.0
UI: preferences moved from separate window to a tab
unclear: Ruby annotation support
https://hacks.mozilla.org/2015/03/ruby-support-in-firefox-developer-edition-38/
unclear: Improved page load times via speculative connection warmup
unclear: BroadcastChannel API implemented
https://hacks.mozilla.org/2015/02/broadcastchannel-api-in-firefox-38/
unclear: Implemented Encrypted Media Extensions (EME) API to support
encrypted HTML5 video/audio playback (Windows Vista or later only)
https://support.mozilla.org/kb/enable-drm
wtf: Automatically download Adobe Primetime Content Decryption Module (CDM)
for DRM playback through EME (Windows Vista or later only)
https://support.mozilla.org/kb/enable-drm
unclear: WebRTC now has multistream and renegotiation support
present for hackers: Implemented DOM3 Events KeyboardEvent.code
37.0.2
several fixes
37.0.1
disabled Disabled HTTP/2 AltSvc
37.0
phone home & UI: pop-up request for rating ("Heartbeat", feedback system)
https://wiki.mozilla.org/Advocacy/heartbeat
disable: pref ("browser.selfsupport.url", "")
phone home: OneCRL centralized certificate revocation
Compatibility: Opportunistically encrypt HTTP traffic where the server supports HTTP/2 AltSvc
https://bitsup.blogspot.com/2015/03/opportunistic-encryption-for-firefox.html
UI: Yandex set as default search provider for the Turkish locale
36.0.4
one security fix
36.0.3
one security fix
36.0.1
several fixes
36.0
Compatibility: No longer accept insecure RC4 ciphers whenever possible
Compatibility: Phasing out Certificates with 1024-bit RSA Keys
Compatibility: some addons might fail
https://blog.mozilla.org/addons/2015/01/13/compatibility-for-firefox-36/
UI: For users who removed the Share & Hello buttons, this new version brings them back unexpectedly
(https://bugzilla.mozilla.org/show_bug.cgi?id=1136300)
35.0.1
lots of bug fixes
35.0
Feature: Firefox Hello with new ???rooms-based conversations model???
Feature: Access the Firefox Marketplace from the Tools menu and optional toolbar button
UI: "New search UI improved and enabled for more locales", whatever that means
UI & Compatibility: Plugin-Finder-Service removed
Firefox will not offer to install missing plugins, for example the Flash player, if a web page needs it.
MISSING IN THE RELEASE NOTES!
34.0.5
UI: Default search engine changed to Yahoo! for North America
34.0
Feature: Firefox ???Hello??? real-time communication client
disable: loop.enabled = false
UI: Default search engine changed to Yandex for Belarusian, Kazakh, and Russian locales
UI: "Improved search bar (en-US only)", whatever that means
Compatibility: Disabled SSLv3
ESR 31
See also this blog posting (includes Thunderbird changes).
Why so many new features in an ESR version?!?
on the right side of the address bar.ESR 24

ESR 17
ESR 10