yes the drivers on that version where different, but we needed to switch to the versions from "debian-buster-backports" instead of "sid" for compatibility reasons, so on 3.8.2 it was a bit newer, for some reason it didn't worked for you the other version (but it worked for me on a machine so the driver itself works, but not with your model)
if you still have 3.8.2 and you boot it, you can see the exact version of the package by running:
❯❯❯ appo nvidia-legacy-340xx-driver
nvidia-legacy-340xx-driver:
Installed: (none)
Candidate: 340.108-2~bpo10+1 <----------- this is the version
so on 3.8.2 you should have a probably newer version
now, if you want to install the drivers on your side (which is very probably), i suggest you to follow the howto: Nvidia Privative Drivers installation but it actually doesn't provide a mention of installing the drivers from SID repos or externally from the official page, so if you are going to need to install them tell me so i can update the howto with the needed info for you 
