First of all, it will come as no surprise to anyone on this thread that I'm going to say that you should use DEXRON-VI!
You asked for an explanation, well the DEXRON-VI additive package is designed to give improved friction characteristics and improved friction durability beyond DEXRON-III performance.
As to the debate about 'synthetic' vs 'mineral', this has been discussed at length on other threads but basically you firstly have to define exactly what is meant by 'synthetic'. The reason for this is that group III formulations can legally be described as synthetic in the USA ... and that's where the confusion arises because some companies describe their DEXRON-VI products as synthetic and some elect not to follow that route.
However, there is also a myth that 'synthetic' (ie. PAO/Ester) products are somehow superior and about 15-20 years ago that would have been true when the majority of companies were primarily blending ATFs using Group I stocks. That is no longer true. Since it is routinely possible to formulate ATFs based on Group III base stocks that actually out-perform 'synthetic' (PAO/Ester) blends. This is not secret information, it has been published in SAE papers and is fully provable.
It is, of course, still equally possible to create formulations that are grossly inferior to 'synthetics' but then again, I've seen companies who have the 'skill' to produce full synthetics which are inferior to mineral formulations, believe it or not!
So, to recap, not all synthetics are equal ~ some are great, some are not and you shouldn't condemn a mineral formulation based on Group III stocks because they can actually out-perform certain synthetics sometimes by several orders of magnitude (depending on the type of oxidation test that it is subjected to). In addition, the DEXRON-VI type additive system is massively superior to the type of additive systems used and approved under the DEXRON-IIIH license (for instance, in the area of friction durability). Of course, some people are going to say that their fluid exceeds the DEXRON-IIIH approval anyway, and therefore is equal to DEXRON-VI, but this is just 'smoke and mirrors' because there's no proof.