Q:

Voltage Equation in Time Domain

When we describe a AC signal (voltage is this case) in time domain we use an equation something like this below: 
\$v(t) = V_m \mathrm{sin}(\omega  + \phi)\$, where  is the angular frequency of the voltage.
I know  is the symbol for angular frequency which is equal to \$2  \pi f\$ and is measured in rad/s.
So why does the voltage equation above show the angular frequency as  and not just ? Is the \$t\$ there just to show the measurement is with respect to time? 
Or is it because  and  not the same?

A:

 and  are the same units (phase in Rads) except  is constant and  expresses the exact phase as a function of time, t for v(t) at freq=

A:

No, \$\omega\$ and \$\omega t\$ and are not the same, and they don't have the same units. \$\omega t\$ has units of radians, which makes it possible to add the phase angle \$\phi\$ to it and get something sensible.
In the equation you provide, the values of \$V_m\$, \$\omega\$, and \$\phi\$ are constants. So, if we want to find the voltage as a function of time, then there must be a variable in the equation that represents the specific value of time in question. That variable is \$t\$.

