They have slightly different usecases. I used them both extensively and one is intended for adding a sidebar style chrome extension ala Streak, while Gmail.js is a simpler event based system.
I personally found both to be lacking in different areas, too bad they weren't merged together to cover a broader base.
Founder of Streak here (we make the InboxSDK). Would love to know which use cases we don't cover but do agree, both libraries were targeted at different layers.
Gmail.js lets you avoid doibng the DOM hacking yourself but is still fairly low level. The InboxSDK was targeted for people who are building apps inside of Gmail. We assume multiple extensions to be running and make sure they play nice, we handle auto updates to handle changes in Gmail and we try to guide developers to using standard Gmail UI styles. We have inbox support coming soon which will let you write your app once and have it work in Gmail and Inbox.
I had more clarity in my thoughts regarding this but it's been a year since I worked on the chrome extension and the startup folded. Thanks for your contributions to OSS.
