What practical, objective differences are there between Emacs and Vim? For example, what can be done using one but not the other (or done more easily with one than the other)? What should I know in order to choose one to learn?
(the text below is my opinion, it should not be taken as fact or an insult)
With Emacs you are expected to have it open 24/7 and live inside the program, almost everything you do can be done from there. You write your own extensions, use it for note-taking, organization, games, programming, shell access, file access, listening to music, web browsing. It takes weeks and weeks till you will be happy with it and then you will learn new stuff all the time. You will be annoyed when you don't have access to it and constantly change your config. You won't be able to use other peoples emacs versions easily and it won't just be installed. It uses Lisp, which is great. You can make it into anything you want it to be. (anything, at all)
With Vim, it's almost always pre-installed. It's fast. You open up a file do a quick edit and then quit. You can work with the basic setup if you are on someone else's machine. It's not quite so editable, but it's still far better than most text editors. It recognizes that most of the time you are reading/editing not typing and makes that portion faster. You don't suffer from emacs pinkie. It's not so infuriating. It's easier to learn.
Even though I use Emacs all day every day (and love it) unless you intend to spend a lot of time in the program you choose I would pick vim