The trapped chest has the texture of the regular chest in the inventory.
How to reproduce:
1. Change the texture of the trapped chest to something very different than the regular chest
2. Take a look at it in your inventory
3. It still looks the same
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is duplicated by
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Affects 13w11a.
Confirmed for 1.5; this is rather annoying in texture packs which make the trapped chest look like an elegant/decorative chest.

Confirmed for 1.5.1!
Very annoying when using texturepacks ...
Uploaded screenshots and test skins for comparison.
I think MC-10757 describes the problem better, it's not only the inventory, hand-held trapped chest items are wrong too!
From my observations, this bug effects trapped chests and anvils in:
The inventory
The hands of mobs and players
Item frames
Dropped items on the ground
As a helmet of a player/mob (requires NBT editing, but Mojang-supported feature)
As falling blocks (requires NBT editing for chests, but Mojang-supported feature)
Inside minecarts (requires NBT editing, but Mojang-supported feature)
All of these situations use the Item Render of the block - making the Item Render display it correctly should fix this bug in all situations.

Confirmed for 13w16a !

Confirmed for 1.5.2 and 13w17a !

Confirmed for 13w18a! (EDIT: and 13w18b+c!)

Confirmed for 13w19a !
I really don't see how this is a bug. They're obviously supposed to look identical.
If they use a different texture when placed, why shouldn't they use the different texture in your hand?
Because they don't. Have you looked at the default texture? They're supposed to look identical. Just because the trapped chest may or may not use a different place in the texture, doesn't mean they're not supposed to be identical. A trapped chest wouldn't really be trapped otherwise.
In fact... the differently looking trapped chest is what appears to be a texture pack problem (MC-10299).
Have you looked at the default texture? The Trapped chest in the vanilla texturepack has a distinct red marking around the latch when placed, but not when in the inventory.
Are you going to say that cracked anvils are meant to have the same texture as undamaged ones, because they looked the same in the inventory? It's the other way around, and they fixed the bug for anvils - I don't see why they shouldn't fix it here.
EDIT: Also, I should mention that not all trapped chests are for traps (there are plenty of legitimate non-trap uses for them), and not all non-trapped chests are safe (thanks to comparators, any and every chest can be trapped).
I'm well aware that there's a small difference on the latch. You can't see that tiny difference in your inventory even if it had different textures in it, so besides custom textures packs specifically made to separate the two, there's no issue. I'm also quite well aware that trapped chests have other uses as I've just placed about three dozen of them for compact storage. That doesn't change the fact that it's meant to be a trapped chest. The anvil is a separate issue, and a very minor one at that.
Note, however, that the difference is very noticeable when you put a trapped chest in an item frame. If the intention was for trapped chests to look identical in all situations, Mojang would not have created two new textures with a unique appearance, and an entire new folder, for them. And minor issue is not equivalent to non-issue; minor issues like these still have some reason to be fixed.
Yeah, and texture pack makers can use this to make alternate chest styles.
I've put a chest and a trapped chest into frames, run on the default texture and there's no visible difference whatsoever.

Still not fixed in 13w21a !

Confirmed for 13w22a !

Confirmed for 13w23a/b !

Confirmed for 13w24a/b !

Confirmed for 13w25a/b/c !

Confirmed for 13w26a !

Confirmed for 1.6-pre !

Confirmed for 1.6.1 !

Thanks for fixing this Jeb 🙂
Confirmed.