Well, it looks like Hugh Hadner made the wet suit and a swedish company made the modern drysuit. Here is what I found on it:
• “SHELL” DRYSUITS
Shell dry suits are defined as any diving suit that keeps the diver from getting wet by a means of a thin waterproof membrane material. The membrane can be single or a composite.
The first dry suits were developed in England in the early 1800’s to work with the newly developed miner’s copper fire fighting helmet invented by Dean, to allow it to be used underwater. In fact the first models were simple open bottom skirts, attached to the copper hat’s “breast plate” by rivets and waterproofed by tar. Thereafter the term “dress” has been applied to the copper helmet “hard hat” or “deep sea” dry suits. These first dry suits were at first made of tarred cotton fabric, but shortly a method of coating natural rubber between two layers of cotton twill was devised. Dresses were made this way until 1964 for the Navy MK-V gear.
Shortly before Word War II, both England and Italy developed dry suits made of sheet vulcanized rubber for their “frogmen” diver swimmers. These dry suits were donned through a front or back “tunnel”.
After WW II, a rubber company in Norway developed a dry suit made of rubber laminated onto a lightweight fabric for hard hat divers. This suit later developed into the “Viking” brand dry suits.
Shortly thereafter a Japanese company upgraded their cotton-rubber-cotton type dresses to nylon-rubber-nylon, which made a very durable hard hat dress for working divers.
About 25 years ago an English firm developed a method of bonding a thick enough polyurethane film to one side of nylon fabric to make it waterproof (“Pack Cloth”). They also developed a tape film that could be hot air welded over the sewn seams, sealing them to fabricate a dry suit.
The latest type shell dry suit is a “re-invention of the wheel” type material known as tri-laminate. This “Tri-lam” uses modern materials, such as nylon or polyester fabric with a modern synthetic rubber, as a water proof barrier between two sheets of fabric, the same way the first dry suit material was fabricated.
• “NEOPRENE” DRYSUITS
In WW II, while plans were being made for the invasion of Normandy, the Navy found that their beach clearance “Frog men” UDT divers would need thermal protection in the cold waters of the English Channel. The conventional “frog man” rubber suits of other navies was not up to the tasks of the UDT swimmers. A professor at a California university was hired to do some tests with a neoprene foam sheet material made by the Rubatex Corporation (they bought the technology in the 1930’s from Germany).
This professor invented the “wet suit”.
In the 1950’s, this material and technology became available to the new sport of SCUBA diving. In the early 1960’s, Rubatex became successful in laminating nylon jersey four way stretch knit fabric to the foam neoprene sheets of the G-231-N high grade material to give the sheets more exterior strength.
The invention of the first “air tight” zipper for early space suits by an English company made the last technical step needed to allow a dry suit to be fabricated from neoprene foam sheet material. A Swedish company developed the dry suit inflation and deflation (exhaust) valves and brought out the Poseidon UNI-suit with a watertight entry zipper that started at the back of the neck and went down through the crotch and back up to the middle of the front. When the popularity of this dry suit became apparent, many wet suit manufacturers went on to develop their own neoprene dry suits.
Diving Suit History and Information
Wetsuit inventor has died at age 92 
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