SMS Backup & Restore should only be backing up your SMS messages, not your photos or contacts.
On the old Moto G, are the contacts saved to your Google account, or only the local Phone account? If the former, then they should be synced automatically to your Google Contacts in the cloud, and therefore should also automatically sync to the new Nexus 6P. You can go to contacts.google.com on your desktop browser to quickly see if your contacts are in your Google Contacts.
If your contacts are only saved to the local Phone account, it's a little trickier. My advice would be to export the contacts (by opening the Contacts app, tapping Menu, then looking for the Import/Export option) into a .csv or .vcf file, then import them into your Google account. Once they're imported, they will sync automatically to any device that's logged into your Google account. See this section of my guide: http://forums.androidcentral.com/ge...-android-updated-2016-nougat.html#post3707512
The best way to transfer your photos is to upload them all to your Google Photos in the cloud, which will also sync automatically to your new phone: https://support.google.com/photos/answer/6193313?co=GENIE.Platform=Android&hl=en
. If you don't want to use the cloud, then I suggest backing them all up to your computer via USB connection, then copying them to the new phone via USB. The backup is important, because phone storage isn't guaranteed to last forever, and phones also get lost or stolen -- so you don't want precious photos saved only on a mobile device.
How were you trying to backup the Moto G to Google Drive? Are you referring to the Settings>Backup & Restore