There are many good things to come out of doing APIs properly. Unfortunately, there are also many bad things that can come out of doing APIs badly, or with misaligned expectations. It is easy to focus on the direct benefits of doing APIs like making data resources available to partners, or maybe developing a mobile application. I prefer looking for the more indirect benefits, which are more human, more than they are ever technical.
As I work with different groups on a variety of API definitions and strategies, one very significant part of the process I see, is people being forced to think outside their box. APIs are all about engaging around data, content, and algorithms on the web, with 3rd parties that operate outside your box. You are forced to look up and outward a bit. Not everyone I engage with is fully equipped to do this, for a variety of reasons, but, overall, the API process does make folks just a little more critical than they are with even their own websites.