What are some specific benefits of cryogenic wind tunnels?
Short answer:
A cryogenic tunnel works by taking advantage of the different properties of nitrogen gas at cryogenic temperatures as compared to the properties of air at normal ambient conditions (speed of sound, density and viscosity). This allows testing at an order of magnitude higher Reynolds numbers over non-cryogenic tunnels. Being able to achieve full-scale values of Reynolds numbers removes a significant source of error in the wind-tunnel test results.
Long citation from https://cryogenicsociety.org/resources/cryo_central/wind_tunnels/:

Dr. Robert Kilgore


The development of the cryogenic wind tunnel is one of many significant breakthroughs in both cryogenics and wind-tunnel technology made during the past millennium. Interest in the development of high-speed commercial and military aircraft resulted in a review of problems of flow simulation in transonic wind tunnels during the 1950s and 60s. One of the more serious problems was the inability to test sub-scale models at Reynolds numbers sufficiently near flight values to ensure the usefulness of the wind-tunnel data. Typically, values of test Reynolds number were too low by a factor of 10 or more.


The search for a way to test at full-scale Reynolds number began soon after Wenham built the first wind tunnel in 1870. Margoulis proposed using a heavy gas or cooling the test gas as early as 1920. Margoulis reasoned that a heavy gas, such as carbon dioxide, when cooled to 253 K would increase Reynolds number and reduce drive power requirements for fan-driven wind tunnels. Margoulis and others of his time concluded the moderate benefits resulting from cooling only to 253 K were not worth the effort.


The idea of cooling the test gas lay dormant until 1945 when Smelt studied ways of reducing the size and power requirements of high Reynolds number high-speed tunnels. Smelt’s theoretical study again noted the advantages of using heavy gases and reducing temperatures. The study by Smelt was presumably a case of independent re-invention of a good idea since Smelt did not cite Margoulis.


Cooling of the test gas was again rejected in the 1940s. This time it was rejected because there did not appear to be a practical way to cool a reasonable size wind tunnel and there were some concerns about finding suitable materials for the tunnel structure.


The concept again lay essentially dormant until 1971. We needed a way to increase the Reynolds number capability of the small wind tunnels equipped with magnetic suspension and balance systems. Dr. M. J. Goodyer was working at the NASA Langley Research Center at the time. Goodyer studied the problem and, again independently, suggested the use of either air or nitrogen at cryogenic temperatures. Goodyer and a small group of researchers at NASA Langley quickly recognized several additional advantages of the cryogenic wind tunnel concept. We immediately set out to develop a practical approach to its application. We quickly built and successfully used a small low-speed atmospheric cryogenic tunnel. It first operated at cryogenic temperatures in January of 1972. We used this tunnel, which had a test section of 18 x 28 cm (7 x 11 in.), to prove the validity of the concept as well as to develop the method of cooling the tunnel by the direct injection of liquid nitrogen into the stream.


We then decided to build a relatively small fan-driven transonic cryogenic pressure tunnel. In its original configuration (since changed), the Pilot Transonic Cryogenic Tunnel had a test section of 34 x 34 cm (13.5 x 13.5 in.) and could operate at pressures up to 500 kPa (5 atm). This extended our cryogenic tunnel experience to the pressure and speeds needed for a large high Reynolds number tunnel. The design of the Pilot Transonic Cryogenic Tunnel began in December of 1973. Initial operation was in August 1993. Again, success led to greater things; this time to the decision to build a large cryogenic wind tunnel to meet the testing needs of the United States. The tunnel would be known as the US National Transonic Facility (NTF) and would be built at the NASA Langley Research Center in Hampton VA. Construction of the NTF began in 1975. It became operational in 1982. It has a test section of 2.5 x 2.5 m (8.2 x 8.2 ft) and operates from ambient to cryogenic temperatures at pressures up to 890 kPa (8.8 atm) at Mach numbers up to about 1.2. Cooling of the test gas is by the direct injection of liquid nitrogen into the tunnel circuit at rates up to 1000 pounds per second.


By taking full advantage of the cryogenic concept, the NTF can test at Reynolds numbers of 120 million. This is an order of magnitude increase in Reynolds number capability over non-cryogenic tunnels. Being able to achieve full-scale values of Reynolds number removes a significant source of error in the wind-tunnel test results. The net result is that commercial airplanes are more efficient and combat aircraft are more maneuverable.


We routinely use cryogenic wind tunnels to test at full-scale conditions (including some scaling parameters other than Reynolds number) for such diverse things as solar towers and submarines. Around the world we now have three large Cryogenic Wind Tunnels used for a variety of aerodynamic testing and fundamental research. These include the US NTF, the European Transonic Windtunnel (ETW) and the Kryo Kanal Köln, both located in Köln, Germany. In addition, there are another 20 or so smaller cryogenic wind tunnels used for a wide variety of purposes in nine countries.


The use of cryogenics in wind tunnels may be unique among the various uses of cryogenics. Most people use cryogenic temperatures because they want to take advantage in the change of some property of a material. For example, they want to turn an ordinary conductor into a superconductor. However, a cryogenic tunnel works by taking advantage of the different properties of the nitrogen gas itself at cryogenic temperatures as compared to the properties at normal ambient conditions (speed of sound, density and viscosity).


As used in cryogenic wind tunnels, cryogenic technology is making a major contribution to experimental aerodynamics.


“It is easy to invent a flying machine; more difficult to build one; to make it fly is everything.” — Otto Lilienthal, Pioneer Glider Pilot