In this app, you can back up your notes to your own Google Drive from the menu (≡) at the top left of the Notes screen → "Backup". This feature is free.
The backup is not automatic. Nothing is synced on its own: your data is saved to Google Drive only when you tap "Create backup data" or "Overwrite backup".
If you use the same Google Account, you can also carry your data over to another device such as a tablet. Both devices use the same backup data, so create or overwrite the backup on one device and tap "Restore from backup" on the other.
You need to overwrite and restore backups manually each time, but it's a handy safeguard when you switch phones or if your device breaks or gets lost.