Deleting and recreating an integration causes several issues:
1) It removes the log history of errors that we record (making it harder for our developers to debug).
2) It removes our ability to test things with your actual integration.
3) It removes all task association history of any currently recorded activity.
This last one is MOST important as this means that if you tracked 10 hours against Task A, once you delete the integration (or even unlink the project) that association is GONE.