What does "Entity API" add to the Drupal API, and why would a module depend on it?
Not complete, but here are some advantages that I know of:


Drupal core only provides an EntityController for loading entities. For save, update and delete, you either need to write it yourself or use Entity API.
You you can expose the base properties of your entities wit hook_entity_property_info(), see privatemsg_entity_property_info() for an example. The advantage of this is that rules.module knows your entity properties and you read and write them with rules. And you also get token integration for free.
It can also automatically provide you with an administrative UI for your entities