I think you're confusing APIs with engines. To understand what an API is it's best to use an example. For example you have Microsoft Word, and you would like to make it read a new type of file or would like to export a file in a different format besides text, how can you do that if Word's source code is not available to you? You use an API that was released for that application by the maker of the application. The API basically lets you use parts of the code that could interact with the application. That's why it's called Application Programming Interface, it lets you literally interface with the application you are developing for. But you might think OpenGL is an API also, but there's no single application. It's an interace to the the graphics card driver, you can program the driver by using the functions revealed in the API. But an engine is more like an interface to a variety of APIs that handle physics, graphics, file reading, producing sound etc etc. So it's more like a program rather then a strict collection of function like an API.
Now to answer if it's easier then a pure API, yes and no. An API is more fundamental, that is you can say DrawPrimitive and specify a triangle to draw and it will draw it, or you can say PlaySound() and it will play a sound, so it's very fundamental and that's what makes APIs easy. The difficulty comes from organizing these very basic calls to create a whole game, as you can see it doesn't handle anything for you at all and you must program all the functionality yourself using these basic building blocks like legos. An engine on the other hand is harder because you have to do some setup that might be less intuitive, like creating scene managers, placing a sound and adding it to your scene before you can play it, creating a camera and attatching it to your player before you can even draw anything etc. You get the idea, you have to have a sense of the stucture of the engine to use it successfully. But it is easier once you get over that initial step because all the code for the most common game techniques is written for you already, you just have to do some setting up and calling some functions, to get it going, then you can concentrate on the logic of your game instead of worrying in which order you have to specify vertices in you model for it to draw correctly as in an API.
I hope that answers some of your questions.