Answer
They are not the same.
Work Step by Step
The wave speed is the speed at which a wave pulse travels along the length of the string, and this is constant. The speed of a wave on a string, as shown in equation 15.14, depends on the tension and linear mass density.
$$v=\sqrt{\frac{F}{\mu}}$$
The speed of a piece of string varies with time as it oscillates up and down, perpendicular to the string.