How does complex data of an FFT and IQ modulation differ?
If the signal frequency is constant and exactly integer periodic in the FFT width, then the transform basis vector for each FFT result bin is the same as a sinusoidal (complex exponential) IQ down-modulator mixer input that starts with a phase of zero at sample 0.  Thus the atan2() results for a signal within that FFT result bin should be identical.

If you only need one result bin of a DFT, then a complex mixer (complex Goertzel) requires less computation than a full FFT result (assuming equivalent numerical precision, etc.).  If you need O(logN) DFT result bins or more, then the FFT is more efficient.

If the frequency is not known to be exactly integer periodic in the DFT/FFT width, then an FFT might be useful as part of an initial frequency estimator algorithm.