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@Mithoron but he also wore the ring during the Battle of Five Armies, and outside of the Goblin cave when escaping from them. I also understand LoTR and The Hobbit not being completely continuous, but the book was revised to bring it in line with the LoTR,, so I would have guessed that what one saw when they put on the ring would have been changed as well.
@Mixxiphoid the accepted answer kinda gives credence to my idea of Bilbo not seeing anything different because Sauron wasn't as powerful (and not looking for him), but it'd be nice to have something supporting the idea, because the only thing confirming it is "when Bilbo found the ring, Sauron wasn't powerful and also the White Council had driven Sauron from Mirkwood but when Frodo had it, Sauron knew it had been found and was looking for it"