We have seen that in Game of Thrones Season 4, Melisandre did not wear the necklace while in a bathtub, and she is looking normal. But in GoT S06E01 she takes off the necklace to reveal her true identity
as a very old woman. Is it that she aged that much between these 2 seasons?
Is it a mistake in the show runners' part or is there any other reason why she didn't appear like that without the necklace earlier?
The necklace is apparently not Melisandre's only source of power, and the fact that we got to see a shot of a vial of who knows what, leads me to think that maybe the bottle is even more important than the necklace.
Since everyone else is speculating, I would speculate with this:
She uses potions to hide her true self. I'm pretty sure her necklace can be filled with the miracle potion, meaning that she doesn't need to ingest any potions when she wears it. See the necklace as a intravenous solution that injects her her needed dose. This might be wrong, but just like all the other answers, we won't know for sure until they address her skills and her potions later in the season.
My interpretation is that she was feeling kind of safe there or simply wanted to rest for once. She undressed. Taking off the necklace and taking off the fake appearance could simply be parts of witches casual undressing routine.
I mean your "take off necklace to reveal her true identity" is just a speculation. Maybe she doesn't have to perform any observable magix stick wavings to change appearance similarly as faces of faceless men can change. It wasn't said anywhere, that taking off the necklace causes any appearance changes and she chose to keep her true appearance hidden when bathing, but not this time.