Until the mid 1970's, the diaphragm design was the standard. This was due to the fact that the first serious US Divers single hose regulators, called Conshelf after the Frenchman's UW habitat experiment, incorporated a "balanced" diaphragm first stage based on the first stage of the two hose Royal Aquamaster. This first stage is still USD's workhorse and some parts are still interchangeable with the two hose reg. While advantages over the piston reg are not as pronounced as heretofore, the diaphragm reg's reliability in salt water environments is still recognized due to its sealed mechanism. One small distinction, the diaphragm first stage intermediate pressure is instantly adjustable which can sometimes be useful.
There was a serious challenge to the diaphragm reg in the '70's. This was due to clever marketing and promotion of another design, the Italian, SCUBAPRO piston design. The stated advantage was that the piston regulator was easier for the average diver to disassemble and maintain in the field. Indeed, this was true as the thing could be field stripped in a matter of minutes. The diaphragm, by contrast, has many small parts that can be tricky to align and install. Moreover, the diaphragm first stage poppet was more delicate than the piston reg's teflon hp seat. On the other hand, the piston reg's O rings were problematical and had a reputation as "leakers". The pistons required very precise alignment during manufacture which some companies had difficulty with. Exposure of the internal parts to salt water and stress of the newly developed 3000 psi tanks seemed to adversely affect reliability. This is one of the reasons, I think, that the piston regs had mandatory yearly shop overhaul, eg to control durability problems. I suspect that Scubapro's refusal to sell aluminum tanks was related to this.
In today's world, both types of regulators have seen extensive development. US Divers diaphragm regs now use redesigned hp poppets whose seats are made of synthetic material and are more durable. Scubapro now uses O rings which are carefully spec'ed for hardness and resilience and are made of space age compounds.(No substitutions, please). Both types have undergone other, incremental changes to the good.
As it stands now, there is no compelling reason to select one over the other, just personal preference. The days of field stripping are just about gone, both types are so reliable, and dive shops are everywhere, it is hard to imagine that this would be a major consideration. Neither delivers "more" air than the other, both types deliver all the air anyone could want. Scubapro used to tout the large bore piston but did not like to be reminded that the piston bore was larger than that of the tank valve, the limiting factor. I'm not certain a case can be made for "sealed" vs "open" or flooded design. I suppose ice divers and others will have preferences but modern piston regs can be sealed if necessary. So, as often happens, there is no immediate gratification, only grist for the mill.
PS, When I changed over to single hose in the mid "70's I bought a Conshelf first stage and mated it to a Scubapro adjustable second. Eventually, as the piston design became more reliable, I bought an "all Scubapro" reg. Currently, I use a MK20/G250hp. I am an open water diver who mostly freedives and spearfishes. However, I never leave the dock without an aqualung aboard.