Deciding when to use the Author Site vs. the Author API really depends on your core product needs, and resources.
How do I figure out the best option?
Our Technical Use Cases section is a great place to start - it provides example client implementations, and spells out which solution is suited to each use case. As a rule of thumb, if you don't have a Content Management System (CMS), and no intentions to build one, then the Author Site is your best route.
Equally, if teacher authoring is core to your platforms offering, then using the Author API is key to achieving this use case.