What difference does it make with APU/Engine air via pack being supplied compared with an external condtioned air source?
The obvious answer is the APU will supply ample airflow to run all three packs and control (reduce!) the temp. in the cabin with a high OAT. An engine running at idle should be able to run a couple of packs reasonably efficiently, but you've got the added problem of safety on the ramp. The 'ground conditioned' air does seem to struggle especially on a 744.
Interestingly enough the AMM states that even with a fully serviceable equipment cooling system, supplemental cooling (pack or ground cart) should be provided if the OAT is >115f, and with one fan off the requirement drops to an OAT of >85f.
The cabin temperature is significantly higher than the OAT especially with all the IFE/lighting/galleys etc, and as the equipment cooling supply comes from cabin air it is a good idea to get some efficient cabin cooling running.