As you said yourself, they both are unit testing tools (although PHPUnit can do other types of testing as well while PHPSpec is limited to unit testing only). However, PHPSpec focuses a lot more on a better code design.
If you are writing tests during development, PHPSpec will let you avoid possible design flaws since it doesn't work with static methods and partial mocks. PHPUnit on the other hand doesn't care how you structure your code, since it only needs to verify that something works as expected. That might lead to suboptimal design and not always following best OOP practices.
Personally I use PHPSpec most of the time and absolutely love it (PHPUnit is a little too verbose for my liking).
For API I'd probably use PHPSpec and Codeception with it's REST module but the choice should ultimately be about your goal and preferred workflow - if you don't follow test-first approach, then you might find PHPUnit better suited to your needs.
