The primary difference is how authorization is handled. In Postgraphile, you map your application roles to Postgres users and define policies using Postgres's RLS. With Hasura, you define rules on roles (similar to Postgres's RLS) but it integrates into your existing auth system using webhooks.
Also, peripherally, we have quite a bit of tooling with an admin UI and a rails-like migration system to make the dev workflows easier :)
Yet, I spent 15 minutes searching for example of using Hasura with an existing schema. Your "Getting Started" page has two examples: Deploy on Heroku and Run with Docker - in both cases, using a fresh database, not an existing one.
EDIT: Docs updated, we'll keep improving it too :)
Homepage has the GitLab example, but not actual "Getting Started" page.