Electric sockets & plugs - role of the live wire

It seems wrong. How can a wire carry a potential difference? Surely wires carry current not pd?

What they are referring to is that the live, or more properly called ungrounded, circuit conductor carries the potential difference to the neutral, or more properly called the grounded circuit conductor, which is intended to be kept at ground or zero potential along with the safety wire, or more properly called the equipment grounding conductor.

See the diagram below for a US residential 120/240 volt rms system, showing a 120 outlet, and the associated wave form. The neutral and grounding circuit conductor is at (or near) zero potential. The live conductor potential varies sinusoidally about ground potential.

What does it mean to "carry potential difference" though? Carrying current makes sense - the wire facilitates the flow of electrons - but how pd is "carried" is not too obvious to me?

I agree it's a poor term to use, and is not obvious. In any event, it is only on the live (ungrounded) conductor where the potential varies sinusoidally with respect to the neutral which is at zero potential. In my opinion, that's the only thing I believe they meant.

Hope this helps.