A pole lathe is a foot powered lathe that allows to turn objects. A cord is wound around the workpiece and turns reciprocally. The pole or bungee retract the cord. One can only cut reasonably on the push.
A treadle lathe usually refers to a foot powered lathe with a flywheel as opposed to one with retracting mechanism. Treadle lathes are not reciprocal, but continuously turn the object. They don't need a cord, but usually have a belt to transfer the movement to the head stock. They are more involved to construct.
Regular carbon steel tools can be used, but HSS tools work equally well. There are pole lathe specific tools like turning hooks that are primarily used in bowl turning (alternating grain).