What is the theoretic difference in the ways to compute the spectrogram?
Firstly $$abs(z)^{2}=Re \left \{ z \right \}^{2}+Im \left \{z  \right \}^{2}=z*z^{^{'}}$$

These are all mathematically identical, so either one will work. 

That leaves the question of whether the square should be applied or not. In nearly all cases the original signal will represent a "linear" quantity such as voltage, current, force, pressure, particle velocity, volume velocity, etc. The physical power is always related to the square (or more precise the product) of these quantities. So if you are interested in actual physical power (or energy or intensity), then you to need to apply the square. That is indeed to convention for audio signals.

[Geek Mode on]: One caveat is that simply squaring a linear quantity makes an implicit assumption about the impedance which can be misleading. For example, if you measure with a microphone reasonably far away from a loudspeaker the acoustic intensity (at this spot) is indeed proportional to the square of the measured pressure. Close up to the speaker that's not the case any more since particle velocity and pressure are not in phase. [Geek Mode off]