P.S: I'm a avid Vim user and I have written couple of extensions for it. It may be easier to understand if the answer provides analogies to Vim and Vim script.
Vim scripts like Chromium and Firefox plugins are inferior to their host processes. They can only modify the application behavior through a set of (supposedly) safe api calls. Since Elisp packages aren't second class citizens, like Vim scripts, it changes everything.
There are other extensible editors, but the line between Emacs' kernel and its extensions is drawn at a different place in Emacs than in any other popular editor. It is more likely that any given feature you're interacting with in Emacs is provided by its extension code than it would be in another editor.
Probably more important is the benefit Emacs gets from being one of the two major Unix editor projects. It has an immense community and user base.
One particular feature that makes it really great for emacs (and a lot of other applications), is that you can define and COMPILE new functions, or even REDEFINE old functions.
Nowadays a lot of dynamic languages have followed the lisp model. They didn't exist when GNU emacs development started.
In the context of emacs, it allows you to edit a new command, and have the lisp in emacs evaluate it (define it) and compile it, hence integrating into emacs code, and making it available for further use. Or indeed, since most of emacs is also implemented in lisp, you can redefine any emacs function the same way, and have it taken into account immediately.
So basically, you are not "extending" emacs so much as you are just modifying it.
Also, lisp has alot of nice and fun features, like lisp macros, first class functions (and now closures), allowing for high order functions, which let you create abstractions easily when programming. Contrarily to most other custom application specific scripting language, lisp is a real algorithmic programming language without artificial limitations.
Emacs and Eclipse have an architecture that allows the creation of applications on top of them that have little to do with what one would call "their base configurations". They allow vertical adaptation to new purposes. Vim is more or less limited to extension horizontally and that's in keeping with it's philosophy of trying to do only one thing and a lightweight footprint.
It's not all sunshine and rainbows. The architecture of Emacs has a dependency on eLisp and my understanding is the dependency is one of the things that has inhibited a move to multi-threading. That processor power has mostly kept the issue at bay, speaks to Emacs philosophy and these days for all its bloat, Emacs would have to be classified as relatively lightweight: the Windows install for 24.5 is a 50mb zip.
For me, org-mode inside emacs makes it more powerful. Org-mode and other extensions are written in Lisp.