sudo apt install firefox
updates Firefox from Linux Mint's servers and to be precise, it either takes one minute or one full hour, but you can't predict in advance which speed are you getting today. This is on a live system, my trusty learning environment for the moment. Meanwhile sudo apt install firefox-esr
downloads the same thing from Debian's servers and never takes more than a minute. This is my experience. It places a different install of Firefox the start menu called Firefox ESR next to the default install called Firefox Web Browser.According to an old discussion I've found which is likely no longer relevant, Debian does not host a Firefox repository, it only hosts an Iceweasel (Debian branded version of Firefox) repository. This is no longer the case:
So Linux Mint doesn't need to host a separate Firefox repository anymore especially if for whatever reason the fast download of updates cannot be guaranteed (I understand it's free software and budgets are limited, but it may be a configuration error as well) as Debian and Mozilla buried the hatchet four years ago. Since more people use Debian than LMDE (so they are either more mindful of their infrastructure or maybe they happen to have wealthier sponsors ), downloads from Debian's servers are always fast while downloading from Linux Mint's servers is either similarly fast or very slow, but the point is you can't predict which end of the stick are you getting today. At least in my experience. Maybe you want to change the defaults.Firefox issue resolution
Debian users who performed a dist-upgrade in June 2016 received the following system message. According to Chris Hoffman of PC World: After a decade, Debian and Mozilla are burying the hatchet. Iceweasel is about to re-assume its proper name.[29]
iceweasel (45.0esr-1) unstable; urgency=medium
* The iceweasel package was replaced with the firefox-esr package.
* Preferences under /etc/iceweasel/prefs will need to be copied manually
to /etc/firefox-esr.
* Other customizations under /etc/iceweasel will need additional manual
steps, through CCK2 or addons.