For some, confusion stems around package managers for the front-end, all vying for developer love. In the field there are three well-known package managers that stand; Bower, Component and NPM (Node Package Manager).
Bower and Component are package managers strictly for the front-end, although Node modules exist to bring them in and use them as Node modules.
The third, NPM, started out as Node only modules, but through the community started going in all kinds of directions (including front-end tooling)
Two different approaches, but the same end goal (sorta): Bower and Browserify.
When it comes to a Browserify package, it needs to be written in CommonJS format and then added to NPM. While the list is high, not all libraries are written in this format which means if you want to bring a dependency in, you need to rewrite it.
Strictly a front-end only package manager. Extremely flexible in that it allows you to install Github repositories as package sources if they are not listed on the official Bower repository.
If the purpose of the framework or library you are wanting to include is mostly for front-end purposes like jQuery or Bootstrap, then sticking with Bower or equivalent is a good idea.
The upside of picking Bower, is that it works really well with task runners like Gulp and Grunt. You are able to achieve the same things as Browserify can (only for client-side libraries) through a few simple tasks you can find in various repositories and gists.
The rule of thumb for picking one or the other is assessing your needs. Are you working on a project that would benefit or needs to work closely with the back-end? The shared code approach Browserify gives you might be best.
If you need more than client-side libraries or like the way Node package work, it is also another added advantage.
If you just need to add a few libraries or a framework to make your front-end all snazzy and move a few elements around the page, style the scrollbars or whatever it is you are doing that requires heavy DOM manipulation and interaction, choose Bower. Using Browserify + NPM for client-side libraries feels dirty and wrong in my opinion.
It is also worth noting that it is acceptable to use both Bower and Browserify in the same project. Using Bower for components with a mixture of CSS and other static assets, then using Browserify for the heavier JS focused code.