Wither Skeleton doesn't always target the player for a while even if he is next to him. Few second after the Wither will begin to attack the player.
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They don't know you're there.
Well they know I'm there, just a delay, as with the other mobs. How long did you wait for it until it attacked you? I made a separate issue about this, and in your description you said it did attack you after a few seconds, so why did you say it didn't target you now?
I'm on Peaceful. I don't know.
Is this still a concern in the current Minecraft version 14w21b / Launcher version 1.4.4 or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
Confirmed for 14w27b.
Seems to only happen in the nether.
Happened for about 7 seconds on average.
I think one thing that causes it is if they can't get to the player, or if they switched from creative to survival.
Happens with nearly every single mob in the game, even when the player is directly in front of them. Tested to happen with all hostile mobs except slimes, magma cubes, ghasts and the ender dragon. I made another ticket concerning this issue that is yet to be confirmed.

Duplicate of MC-48532.
I've seen this happen with all mobs from the latest snapshot, they are all a bit buggy.