You basically get a unification of backend and frontend languages -- both using JS, the same dependency and versioning system, the same way to require modules, and of course the same code. Being able to use the same libraries on the backend and frontend is huge. (But not required. Browserify/npm are still useful if you don't use Node.js.)
And it's cleaner. Relying on the Node.js dependency system means you can just import modules the same was in Node.js, instead of maintaining an ugly "vendor/" folder where you manually copy things like jQuery.
I suspect (I haven't gotten this far yet) that you can more easily unit-test your code this way. You just need a "headless" browser environment (using jsdom etc.) that's compatible jQuery.
Browserify is actually very small tool. It leaves all package management to npm, and all Browserify does is to generate a JS file that includes all your package.json-specified modules in neatly namedspaced dependency-initialization order, ready to be imported using a script tag.
There is also Bower [2] from Twitter. Unlike Browserify, Bower is mainly a wrapper around npm that acts as a separate package manager, but doesn't actually implement the glue needed to import any JS on the client side.
[2] http://bower.io