API Routes and Server Actions can indeed be used in similar ways: ultimately, they both allow you to write server-side code that can be called from the client-side. However, let's look at the use-cases for each of them to understand the differences.
TLDR
Use API Routes when fetching data, and Server Actions when performing actions/mutations.
API Routes
API Routes are more generic and a lower-level abstraction than Server Actions. They are a way to write server-side code that can be called from the client-side. They are a great way to expose a REST API, or to write server-side code that can be called from the client-side.
The downside of API Routes is that they need to be called using the Fetch API, which you can use directly, or through a library such as React Query or SWR. This kit uses SWR.
Generally speaking, API Routes should be preferred when fetching data - rather than performing actions/mutations.
Server Actions
Unlike API Routes, Server Actions are a higher-level abstraction. They are a way to write server-side code that can be called from the client-side as POST requests.
They are particularly useful when you want to perform actions/mutations on the server-side, such as creating a new user, or updating a user's profile.
Thanks to utilities such as revalidatePath and revalidateTag, you can also use Server Actions to invalidate the cache of a specific page or tag (used with fetch).