Yes, they are basically the same idea but there are huge differences also. In a phasor representation we try to represent a wave by its magnitude and phase, which essentially translates into a complex number in an Argand plane. So 100 Exp[-i w t] would be a vector having magnitude 100 and phase (- w t). This is particularly helpful in linear systems, where the frequency of operation will be unchanged irrespective of the processing done to the signals. So usually phasor analysis is done if signals are composed of a single frequency component, essentially a sinusoid. And if the system is linear, then the frequency of the sinusoid does not convey any information, because its going to remain constant. The only things that changes is the magnitude (voltage shift) and delay (time shift). Also in wave-guides and optical elements, we can use this concept only if the properties of the medium (i.e, permittivity, permeability, refractive index remain constant), because if they change then frequency will change, just like light changes its frequency when penetrating from air to water. So the basic disadvantages is that when we have signals composed of multiple frequencies (remember Fourier series expansion), or when the system under study are non linear, then we need to consider multiple frequencies onto the same Argand plane, which increase the complexity of analysis. What Fourier did to do away with this problem is that he placed many Argand plane stacked on top of each other. Each plane corresponds to a single frequency only. Now you draw all the phasors on their respective planes. Then look at the stack from the TOP, giving you the Top View or projection of the phasors in the real axis and it should be a continuous curve (most of the cases). Similarly, SIDE View will give you imaginary part. Draw what you see on two separate graphs. They will give you Re{} vs Frequency plot and Im{} vs Frequency plot. Now we plot another two graphs, which are
1. Sqrt[ Re{}^2 + Im{}^2 ] vs Frequency <-- Magnitude spectrum
2. ArcTan[ Im{} / Re{} ] vs Frequency <-- Phase Spectrum
So we can imagine a frequency spectrum as a Top view and a Side View of a Three Dimensional Curve. The three dimensions were Re{}, Im{} and Frequency.
So Frequency domain is Superset of Phasors !!! Nothing Else.