I'm semi-competent with vi keybindings, but I prefer Emacs overall. The reason these editors have such fervent adherents is because the editing model they provide is more powerful than newer systems, which is why providing "vi keybindings" or "emacs keybindings" isn't enough, even if you aren't using any extension features or customizations for emacs or vi.
I'm only going to talk about Emacs' model because I understand it best. The common model for text editing today involves a buffer of text, in which text can be inserted, deleted, selected, and cut/copied/pasted to the system clipboard.
Emacs buffers, of course, can support these operations. Along with tracking cursor position for each window they're visible in, they also keep track of "marks" made in them. The text between the "point" (cursor position) and the "mark" is called the "region", and roughly corresponds to the selection in mainstream editors.
The difference is that Emacs keeps track of the last several locations the mark was set at in the mark ring, and you can return to them with a keystroke (or two, depending on your configuration). I find this extremely useful, especially since a lot of Emacs commands that change your location in the buffer set the mark at your old location. An example is when I'm editing a Python module and need to add an import statement to the top of the file. The keystroke for going to the top of the buffer (Alt-<) sets the mark. I add the import statement. I press Ctrl-u Ctrl-Space and I'm back where i started. I can keep doing this to cycle back to previous positions as well. (Maybe I needed to select some text while adding that import statement.)
The other (and more well-known) Emacs difference is the kill ring. Most of the keystrokes for removing text from the buffer save text to the kill ring, which can then be recalled with the "yank" command (Ctrl-y). The essential feature is that subsequent yank commands retrieve older killed text. So you can kill several sections of text in a row, then retrieve them in order. You can also cycle through the kill ring with Alt-y after a yank, removing the retrieved text and inserting the next entry in the ring.
Emacs had these features in 1978. The only other major system to adopt them to any extent is NeXTStep (and now inherited by Cocoa). Other tools provide more features for specific tasks, can be extended in languages way easier to use than Emacs Lisp, and have nicer visual interfaces... but Emacs remains better at text editing. Which is why, once you know how to use it, it's so hard to quit.