Barley twist refers to the appearance of the surface of the barrel and is created by the manufacturing process.
It is widely believed that the Barley twist barrel is/was of a better quality than the current barrels.
From what I've read, this seems to be a result of the Barley twist barrel being generally less pellet fussy than the current one.
But then perception is nine tenths of the truth.
I bought one, tried it, found it be no different to my current barrel.