Cryptids (幻獣, genjuu; 未確認動物, mikakunindoubutsu; UMA, Undefined Mysterious Animal) are animals that scientists claim to be a hoax while natives/cryptozoologists claim to exist or another term would be an animal that has yet to be discovered by science. In Japan, they're more commonly called UMA, similar to how UFO is used.
There is no real difference than a regular animal other than being out of place (habitat limitations) but some may possess some unique traits. They would function and look no different than any other living creature that is known to exist today and if they are ever found to exist then they would be placed within their correct species, genus, etc. Some are animals that used to live before but have gone extinct. Few cryptids were found to exist after scientific study and finds.
Cryptids in Touhou
It is unknown if cryptids like regular animals can become youkai beasts as well. While the extinct yamainu commonly known as the Japanese wolf would be considered cryptids if alive in the outside world, it appears as a common animal that never went extinct in Gensokyo. In the case that the yamainu is a cryptid, then the okuri-inu is the only known cryptid to turn into a youkai beast. However becuase the yamainu actually existed, it would not be a mythological animal.