No, ω and alpha are not the same thing. One is rotational velocity in radians/second, and the other is rotational acceleration in radians/second^2.

For a particle traveling on a circular trajectory of constant radius r there are two components of acceleration. The first is acceleration in the direction acting along the radial vector: a_r = (ω^2)r. The term ω is the angular velocity, expressed in radians per second. This component of acceleration is called "centripetal" acceleration, and from F=ma there is a force required to cause the particle to curve - that force is what you feel when you swing an object tied to a rope in a circle.

The second component of acceleration is the tangential component, a_t = alpha times r. Here alpha is the rate of rotational acceleration in radians per second squared, which causes the particle gain speed. You can think of it as the equivalent of linear acceleration. To make the particle gain speed a force is require to push it, again from F=ma that force acts in the direction of travel of the particle.