I'm using Baubles 1.0.0.16 and Minecraft 1.7.10 (but this also happens in 1.7.2).
Upon starting a new single-player world, I get these errors in the console: http://pastebin.com/jwLcU77f
It's not enough to crash; things still work. But they also come up whenever I open the Baubles inventory. And, sure enough, if I equip a bauble like the Miner's Ring and then quit and reload the level, it's gone.
Is this some kind of permission error where it can't access files at that directory or something? I'm not entirely sure what's going on.

I believe "\saves\New World\players" should be "\saves\New World\playerdata"

This is due to Minecraft changing the foldername in their saves to "playerdata", yes. You can cheat around it by simply creating a folder named "players". Baubles will still save it's info there.

1.0.0.16 as-is isn't compatible with MC 1.7.10, nor does it claim to be. Use Vazkii's version for now, or wait till Baubles 1.0.1.+ which should be out soonish.