Been down this road and here are a few answers,
First, with regards to traction aids, some of the limited slip and locker manufacturers are building units designed before the general availability of syns. These devices are "calibrated" to the friction characteristics of mineral oils. In the case of two limited slips, I was able to talk to engineers at those companies and get the reasons why.
The engineers at Auburn Gear say the cone clutches on their HP or PRo Series LSD (designed by Borg Warner in the '50s) will immediately loose effectiveness (loss of bias ratio) with synthetics and over time, the cone clutch surfaces will become polished and the bias ratio (just google the term if you don't know it) effectively goes away.
The Eaton Truetrac (designed by Tractech and introduced in 1988) is a gear type limited slip that was designed for use with mineral gear oils. Since the unit relies on a certain amount of friction between the gears, a reduction of that tends to reduce the bias ratio. Some syns have been known to cause an odd squeaking noise in the Truetrac when it's biasing. This is why the instruction say "mineral oil only."
Synthetic oils, in general, do not resist moisture as well as a good mineral oil, so axles or gearboxes that face moisture may be better of with a mineral (a 4x4 for example). Hydrolysis (chemistry changes) and corrosion can be more a problem with synthetic gear oils when water is present. It's not universal and some oils are better or worse.
I interviewed the owner and one tech at a well known axle builder on the west coast as to why they recommend mineral oil vs synthetic. I'm not going to name them but it was clear to me they weren't all that knowledgable about oils. They relied more on cause and effect in their narrow slice of the market to dictate their policy. The explanation they gave me, and I'm just repeating what they told me, not endorsing it, is that synthetics left the gear faces of the ring and pinion gears "polished" and down the road, it was about impossible to read a pattern on them. Since they dealt with race and HP cars that were frequently torn down for various reasons (more so than an ordinary car), a polished gear would make it hard for them to set back up correctly.
I've done a fair bit of axle work and gear setup and I haven't seen many "polished" gear faces. I talked to a few other axle builders and their penchant for mineral gear oils seemed even less coherent. On one hand, you have to figure these professionals must know what they are doing. On the other hand, what I know about oils leads me in another direction. I reserve judgement until I get a clearer explanation.
Would I disregard advice to use a mineral? Maybe. I would heed it when the mfr. makes it a part of a warranty, which some of the aftermarket axle builders do. I would heed it in the case of a traction aid because if I install a LSD, I want to get the full benefit from it. Otherwise, I would use a syn for the obvious benefits.