Essentially, Flynn is a self-hosted Heroku. It comes with the same sort of benefits as Heroku, including "git push" to deploy and a quick CLI tool to manage things such as domains and databases in your cluster. The promise of having a single command deploy your application and another to scale it is fantastic.
Though Flynn comes with the same benefits as Heroku, it comes with a lot of drawbacks as well. First, you have to administrate it yourself. This means hiring a dev-ops engineer to manage a cluster of servers and the Flynn platform itself, or sidetracking a developer every time there's an issue or an upgrade to perform. Heroku, though more expensive, is an outsourced version of this.
It's also under pretty heavy development. There's updates fairly frequently, which means you'll be doing a lot of upgrading your cluster. If you have more than 3 machines, you'll be spending a lot of time upgrading and monitoring.