In circular motion we user the Greek letter $\omega$ (omega) to represent angular velocity, so the angle $\theta$ travelled through at time $t$ is
$\theta = \omega t$
Typically $\omega$ is in radians per second, so the time taken to complete one rotation is
$\displaystyle t_{rot} = \frac {2 \pi} {\omega}$
and the reciprocal of this is the frequency of rotation
$\displaystyle f_{rot} = \frac {\omega} {2 \pi} $
When describing linear SHM we use $\omega$ to denote an angular velocity in phase space rather than in real space. So the displacement $x$ at time $t$ is
$x(t) = A \sin (\omega t + \phi)$
where $A$ is the amplitude of the motion and $\phi$ is its phase, so that $x(0) = A \sin (\phi)$. The time taken to complete one oscillation is the time taken to complete one rotation in phase space
$\displaystyle t_{osc} = \frac {2 \pi} {\omega}$
and the frequency of oscillation is the reciprocal of this time:
$\displaystyle f_{osc} = \frac {\omega} {2 \pi}$
With linear SHM, linear velocity is:
$\displaystyle v(t) = \frac {dx}{dt} = A \omega \cos (\omega t + \phi)$
With linear SHM there is no danger of confusion between $x$ and $v$ in physical space and $\omega$ and $\phi$ in phase space. However, with rotational SHM we often re-use the same notation as in linear SHM, but now the left hand side of the equation is angular displacement $\theta$ instead of linear displacement $x$. So we might write
$\theta (t) = A \sin (\omega t + \phi)$
Note that in this expression $\theta$ represents an angle in physical space, whereas $\omega t$ and $\phi$ represent angles in phase space. Confusion may arise if we want to write an expression for angular velocity and we naively write
$\displaystyle \omega (t) = \frac {d \theta}{dt} = A \omega \cos (\omega t + \phi)$
This is incorrect (or, at least, very confusing) because we are using $\omega$ to represent different things on either side of the equation. On the left hand side it represents an angular velocity in physical space, which varies with time, and on the right hand side it represents an angular velocity in phase space, which is constant in SHM.
To avoid this confusion we could either use a different label for the angular velocity in phase space e.g.
$\omega(t) = A \Omega \cos (\Omega t + \phi)$
or we could change the way we denote angular velocity in physical space e.g.
$\dot {\theta}(t) = A \omega \cos (\omega t + \phi)$
