Choosing to use local storage means that your data will be stored in the browser where the widgets are opened, and the configurations from this website will be present in the widget URL. To change the configuration you'll have to change the URL of the widget.
Choosing to use cloud sync means that you will receive a new storage ID to keep your data saved in the cloud, beeing able to reuse it in any browser, at any time. Keep in mind that this ID should be private. Cloud sync is powered by Firebase Firestore.