If your application is something for whom an API is useful, then encapsulate the application behind your API. This will allow you deal with the application at a somewhat abstract level, which will lead to better programming practises. So, write your tentative API first, program for the API, test with rough mock-ups, then write the front end.
I think I agree with the people saying it can be good to develop your API in tandem with your app. However, I think that developing the API first leaves one in danger of being perfectionist and stops one adhering to ancient proverb, "Just fucking ship."
Nah. Let's say you want to retrieve an artist from a query string.
First, define your API for this:
external->getArtist(string)
Now program the getArtist function. Then make a gui that uses this function, deploy it and you have already shipped your first version with a single API function.
