Some of the reasons you'd want to use Bower:

* Your external libraries have dependencies, and having dependencies managed for you is a good thing.

* The external components you're using include both JS and CSS, and need to be served separately.

* Because you love your users, and want to concatenate your javascript and CSS to single files before serving it out to your users.

* Because you're including components that expose SASS mixins you want to use inside your existing code.

The hosted vs serving your own libraries is a complicated topic, but the basic answer comes down to "the less requests your clients have to make, the faster things will be".