Phasor Representation of Sinusoidal Forcing Functions
They are equivalent in that they represent the same sinewave with amplitude \$V_m\$, phase \$\theta\$ and frequency \$\omega\$ (implicit at this level). Mind that, although they are often represented by exponentials, sinusoidal signals are purely real. Therefore you always have to take the real part of the exponential form, and that's how you obtain the cosine.

The phasor helps you in expressing the phase as an exponential, therefore making simpler to operate with different signals and reactive components.