Why is it that some websites (like this one) remember my username and password, but others (Facebook, for instance) only remember the username, despite me having enabled said remembrance in “Tools/Options”?

Firefox?
If you go to “Tools”, “Options” and then “Security”. Click on the “Saved Passwords”. Firefox will have saved all your passwords even though the individual pages don’t. You’ll need to activate remember password on each web page, I think.