Categories aren't just graphs; they have additional properties, the most important of which is composition (ex. function composition). They can also be thought of as a type algebra, like groups or rings, but again with different properties.

Edit: these extra properties allow you to reason about a wide class of problems when they're represented as categories, not just round trip them from unencoded to encoded to unencoded. For example, Pijul uses a category of patches to model file patching and prove certain properties of the version control system.