As the title suggests, which operations does the "sync branch" perform on the github application for OS X? Does it do a full clone of the repo again?
According to this article on GitHub app documentation:
No, SyncBranch doesn't perform a full clone of the repo. It just pull the changes in origin and push all the commits you haven't pushed yet, all in a single step.
See full article content:
How can I push or pull?
If your repository has a remote connected, we encourage you to use the Sync Branch button. In one step, we'll bring in new changes from the remote and push any commits you haven't published yet. Behind the scenes, we do the equivalent of a git pull --rebase (but make sure to never rewrite merges).
If you would like to perform a standard pull or push without syncing everything, you can do so via the Repository menu item.