Consider, a particle is moving in a harmonic oscillator potential : $V=\frac{1}{2}m\omega^2x^2$. The force on the particle will be : $F=-m\omega^2x$.
What is the unit of $\omega$ here ? Is it $Hz$ or $rad\;s^{-1}$ ? From the force equation, it appears that if the unit of $\omega$ is taken as $s^{-1}$ or $Hz$, the unit of force comes as $Newton$, but if it is taken as $rad\;s^{-1}$ it doesn't seem to be coming in $Newton$.
Even though the units for $\omega$ are radians/sec, radians have no physical dimension. In both cases the dimensions will be the same. That is, $\text Hz$ and $\text{rad}\ s^{-1}$ have the same physical dimensions since radians are dimensionless.
In the equation $F=-m\omega^2x$, the dimensional units on the right hand are $\text kg\ m\ s^{-2}$ or physical dimensions $[M][L][T]^{-2}$ which is the dimensions for force, consistent with the units of force, or $\text {Newton}$.
The definition of Hertz is 1/s. Radians or degree's are essentially unitless by convention, therefor your units for the angular frequency would still be in 1/s or Hz.
Look at this for the unit conventions of radians and degrees: https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless
