The update of Iceweasel (the Debian version of Firefox), version 18.104.22.168+dfsg-3 seems to be very unfortunate.
First of all, after the update, when running the new version, the import profile wizard came out and after this, a brand new profile of iceweasel came out, just like I'd lost my previous profile.
Fortunately, my previous firefox (or iceweasel) profile was not lost... simply, a new directory (called, indeed, iceweasel) was created under the .mozilla directory.
So, I only had to remove the new iceweasel directory and make a symbolic link, with the same name, to my previous firefox directory.
Some problems, however, still existed: when clicking on an URL from within Thunderbird (well... yes, OK, Icedove in Debian), if a running instance of iceweasel was present, the present error came out:
Iceweasel is already running, but it is not responding. To open a new window, you must first close the existing Iceweasel process, or restart your system.
Fortunately, the solution is illustrated in this bug report, and simply consists of changing this line in the file /usr/lib/iceweasel/iceweasel:
And then, everything worked as it used to do before...