I am completely confused about SMB3 vs AFP connections, although all the machines (running from new iMac to 2007 mini) will connect with either AFP or AFP over IPTC, or SMB, I can never tell.
As rccharles says, AFP is Apple's original file sharing protocol, and it is required for Time Machine as there are specific features Time Machine needs that Apple has included in the AFP protocol.
transfer rates seem slow, but perception is not reliable and I don't know how to test the transfer speeds.
CIFS/SMB protocols are where Apple is leaning towards (except for Time Machine as mentions above), as CIFS/SMB are a bit more efficient. But it is also important to tell the Finder to NOT show icon previews (slows down CIFS/SMB really bad for large folders)
However, a lot of performance depends on the speed of your network connection (WiFi can be very variable), the performance of the NAS device (slow disks, or under powered processor in the NAS can affect performance).
You can get a utility such as the Blackmagic Disk Speed Test from the App Store
<https://itunes.apple.com/us/app/blackmagic-disk-speed-test/id425264550?mt=12>
Or just find a very large file (at least a gigabyte or larger), and copy that file to or from the NAS of choice, deleting it between copies, and switching to different protocols, different WiFi frequences (2.4GHz vs 5GHz) or via Ethernet, or a different NAS for each test. Keep notes. Repeat a few times to make sure you have consistent results.
I'm also getting a host of permission errors on documents I saved to a network drive when connected via SMB when I later tried to access them through AFP. ???
That is a function of the NAS device and how they are implementing permissions and then converting them for the different protocols. That is to say, the NAS device says yes or no to any access, not your Mac. And if the NAS is not treating the AFP user and the CIFS/SMB user as the same user, or if the NAS is storing permissions separately for AFP and CIFS/SMB, then granted permissions for one protocol would not transfer to the other protocol. So you really need to either use the same protocol to access the files, or figure out why the NAS is issuing permission errors for your access.