If you had not imported the photos to the computer before the update, or had not been using iCloud Photo library to sync, then they are gone. Since it appears that you made a backup with the photos gone and then restored, then you have copied over any backup that may have contained the photos.
For future reference, you should always make a backup just prior to updating a device. Also, it is a good idea to import your photos from the iOS device to the computer for safe keeping, especially if they are as valuable as you say. The iPhone is a temporary storage device, which like a computer, can experience a problem at any time. iCloud Photo library allows for you to upload photos to the cloud and sync them back to your devices. It is not permanent either, as deleting photos from devices will also delete them from iCloud. But since it appears you had iCloud turned off, then they would not be there, and not backing up before the photos disappeared, but backing up afterwards has written over any backup that may have the photos.