John Musser offered a presentation at Web 2.0 Expo titled Open APIs: Big Picture and Best Practices (that's a PDF link).

His slides #4, explains why publishers should care about APIs:

• Ebay uses them to make money: Over 45% of all products get listed via their APIs
• Google Maps uses them to build their brand, seeing 300% growth vs 20% MapQuest
• Salesforce uses them to extend their audience reach: over 50% of all transactions via their API and 500 apps for specific verticals
• Flickr uses them to innovate, with hundreds of apps built to date

And for users, the benefit is far easier to understand: free/cheap functionality. Musser notes that Smugmug used APIs to save money ($500K/year) with Amazon S3 Storage.

Most of this argument applies to Widgets as well - they're just simple presentations of platform APIs, after all.

There's plenty more in the presentation, and Jason Snyder did a nice presentation write up for InfoWorld that warrants a read.