Beyond provide a profile of an instance's stack, what is the purpose of Connections?
It is described a little bit here: https://issues.civicrm.org/jira/browse/CRM-16173

Seems like the intent is a generic framework. I believe this link is a reference implementation of how to write a service that would interact with this framework: https://github.com/civicrm/cxnapp

Maybe others know of some real services that do implement this.