Angular frequency and angular speed are two distinct but related concepts in physics, each measuring different types of motion.

Angular Frequency
Angular frequency, represented by the Greek letter $\omega$, quantifies how quickly a wave oscillates, measured in radians per unit time. It is particularly relevant in the analysis of waves and oscillations, such as sound waves, light waves, and other forms of electromagnetic radiation. Angular frequency is closely related to two other important concepts: the frequency ($f$) of the wave, which indicates the number of oscillations per unit time, and the period ($T$), which is the duration of one complete oscillation. The relationships among these quantities are expressed by the formulas:

and

Angular Speed
In contrast, angular speed, also denoted by $\omega$, measures how rapidly an object rotates around a specific axis. It is defined as the angle through which an object rotates in a given time interval. Angular speed is frequently applied in contexts involving rotational motion, such as the spinning of a wheel, the orbiting of a planet, or the rotation of a top. The relationships between angular speed, the period of rotation, and the frequency of rotations are the same as those for angular frequency:

and

Summary
In summary, both angular frequency and angular speed are measured in radians per unit time and share the same symbol $\omega$. However, they apply to different physical phenomena: angular frequency pertains to the oscillation of waves, while angular speed relates to the rotation of objects. Recognizing the distinction between these two concepts is essential for a comprehensive understanding of physics, especially in the areas of wave mechanics and rotational dynamics.