Why should I use an API as django-tastypie or django-rest-framework?
For the exact same reasons than you used the django-admin for the admin instead of writing views, forms and templates from scratch:

less work
tested and approved by many other developers
therefore more secured
DRF has a really nice architecture. When you use it, you'd think "that's how I would have done it" (the truth is "this is how I wish I would have done it").

DRF also let you browse/test the API via HTML.

'External developers' who wants to use the data from my website on their own website?

Yes

For me to get a nice overview of my data (even though the admin does this job good)?

Admin is better indeed for that.

Use the create, read, update and delete functions in ajax calls on
  different pages on my website?

Yes

Or should all views that get data from a model get the data through an
  API to check permissions etc.?

You don't have to get data through your API (is that what you mean?). I've never setup an application this way. That being said, you can do it, start by building an API both for you and external developers especially if you're going to use a lot of Javascript.
I have been looking at django-tastypie and django-rest-framework.
What is the advantage of using an API?
I've thought of following use cases. Which use cases is an API primarily intended for?

'External developers' who wants to use the data from my website on their own website?
For me to get a nice overview of my data (even though the admin does this job good)?
Use the create, read, update and delete functions in ajax calls on different pages on my website?
Or should all views that get data from a model get the data through an API to check permissions etc.?

I have read some of the documentation for both APIs but it's still not completely clear to me.
Django has request.is_ajax() and Django 1.7 introduced JsonResponse so I cannot see why a big complex framework could make a better job sending and receiving json but I guess I'm wrong based on the number of developers who use the api frameworks :-D