UPDATE: the behavior described below wasn’t intended. pushed_at will be updated any time a commit is pushed to any of the repository’s branches. updated_at will be updated any time the repository object is updated, e.g. when the description or the primary language of the repository is updated. It’s not necessary that a push will update the updated_at attribute — that will only happen if a push triggers an update to the repository object. For example, if the primary language of the repository was Python, and then you pushed lots of JavaScript code — that might change the primary language to JavaScript, which updates the repository object’s language attribute and in turn updates the updated_at attribute. Previously, the primary language was getting updated after every push, even if it didn’t change (which wasn’t intended), so it triggered an update to updated_at.
The difference is that pushed_at represents the date and time of the last commit, whereas the updated_at represents the date and time of the last change the the repository. A change to the repository might be a commit, but it may also be other things, such as changing the description of the repo, creating wiki pages, etc. In other words, commits are a subset of updates, and the pushed_at timestamp will therefore either be the same as the updated_at timestamp, or it will be an earlier timestamp.