By project activity schedule, I'm going to assume you mean a detailed project plan as opposed to a high level document like a project milestone chart. The project milestone chart which just outlines the high level deliverables and their dates is a good communication for the project sponsor and executive management.
On the other hand, documenting the project activity schedule or the detailed project plan can serve two purposes that immediately come to mind. First, its let's everyone on the team know what to do (if its being published to them). Second, it is important project history that may serve future projects well to know in detail how things were done, how long they took, how many resources were involved, and so on. It will help future estimating and planning efforts.