Blasted through the rest of NG+ and started up NG++. I was hoping I could use my Soul Memory or previous Great Souls in my Inventory to bypass the Shrine of Winter and get to the Chancellor so I can polish off the last 3 Achievements, but sadly that was a no go. It wasn't a speedrun, but I did manage to pick up 3/4 Great Souls in a little over an hour.
Thus far NG++ hasn't really seemed different from NG+. I expected more Red Phantoms, but I haven't noticed anything different.