Waffle's integration with github's project management tools (e.g. issues) is definitely what I like best. I love github's issue documentation, and the ability to link to/close out issues through pull requests and commit messages. It makes it very easy to create a living documentation for your app. However, when you want to do project management, especially Agile, using github issues, github's GUI is anything but ideal. By integrating with github, waffle's GUI really improves upon githubs.
I have always felt that the integration with github, while quite good, was somewhat spotty. Someone would move something on a waffle board and the issue tags wouldn't get updated in github, or vice versa. This often left members of the software team frustrated, because we were never quite sure what system to trust. I also disliked the lack of ability to calculate team velocity based on difficulty points assigned to issues.
Consider keeping Github for your documentation, but using Pivotal Tracker for project management. There's some duplication of effort, but it might be worth it if Waffle's connection frustrates you.
The big business problem this helped us solve was how to plan and estimate the end date of software development projects. Waffle helped us to stay organized and be a little better about our projections.