You may be missing the Vista KB2763674 update.

https://support.mozilla.org/en-US/kb/get-latest-version-firefox-windows-xp-vista

That is affecting downloading Firefox 43.0.2 and later for WinXP and Vista, the latter when Vista is lacking the KB2763674 update. WinXP doesn't have that update available.

BLC1456 said

There is a strange phenomena in regard to Firefox 44.0 and Windows Vista. If you want to download the “Firefox Setup 44.0.exe"-file using Firefox from the official homepage (https://www.mozilla.org/en-US/firefox/all/) everything works as usual as long as you use Firefox with administrator rights. But if you use Firefox without administrator rights you are not able to download the current 44.0.exe-file. Instead you can only download the “Firefox Setup 43.0.1.exe". And even then its not possible to update to 44.0 via the build-in update process. Why is it? How is Mozilla able to detect if you are surfing with administrator rights or not? Does Mozilla use any website techniques to do so? This happens even with Javascript deactivated and Java, Flash, etc. not installed. Or is it a kind of “build-in thing” in Firefox?

I wonder if the useragent of the browser you are using while as Admin is different? like saying you are using a newer OS.

If the mozilla.org sites sees you have Windows NT 6.0 which is Vista or Windows NT 5.1 which is WinXP in web browser useragent, then you should get served 43.0.1 instead of 44.0

WinXP users need SP3 to run the 43.0.2 or newer setups. Problem is there is many users still using WinXP SP2 which cannot run the Firefox setups since 43.0.2.