Both have the concept of modules, and they overlap in this area.
Browserify allows you to use Node style modules in the browser.
RequireJS allows you to asynchronously load parts of your app, while Browserify bundles all JS into a single file.
If you're planning on going that route with RequireJS, you should also look into AlmondJS: https://github.com/jrburke/almond
Grunt's grunt-contrib-requirejs[1] does a good job of traversing your dependency graph with r.js and building out a single bundle, but it is not especially quick; it takes about a full 3-5s on my current build. But it's a hell of a lot better than the alternative.
Of course, if I were to start over, I'd run browserify and either build the files in a build step with Grunt or Gulp, or just use browserify-middleware[2] and skip the build step entirely (with something like node-sass for styles).
[1] https://github.com/gruntjs/grunt-contrib-requirejs [2] https://github.com/ForbesLindesay/browserify-middleware
Browser caching is nice, but if you're doing a continuous integration/deployment type, you're probably invalidating your cache often. If you use a single build you may be invalidating all of your JavaScript for each tiny deploy even though maybe 1% of your JavaScript changed. Another reason to want modules.
I don't know if Browserify can do that.
Using the r.js optimiser, we are able to create several modules for our SPA that include logic and templates, and load them at runtime only when required. By default, we load the most used bits of our application, and load the rest when the user accesses the more rarely used views.