The API is an interface to your data on Teamwork.com. The API is used from your server to the Teamwork.com platform and can be used to List, Create, Edit or Delete items.
Webhooks are automated calls from Teamwork.com to your server triggered when a specific event happens in Teamwork. For example, when a task is completed and you want to know about it in real time we'll make a POST request to the URL you have registered for the EVENT.COMPLETED webhook in your Teamwork account.
So, in a nutshell: The API is where you tell us things and Webhooks is where we tell you things.