Discovery of the Difference
Jade was just jade through many generations of the world's prehistorical cultures--from the dynastic Chinese to the Mayans--who used jade as jewelry, medicine and tools and weapons. In 1863, according to Geology.com, jade was further differentiated by French geologist Alexis Damour into jadeite and nephrite.
Jadeite
According to the International Colored Gemstone Assn., jadeite is rarer and different in composition to nephrite. The stones just look and form in similar ways. Jadeite is made of pyroxene formations in metamorphic rocks that are heavy with aluminum, and is slightly harder at 7 Mohs than nephrite's 6.5 Mohs. It is usually glossier-looking than nephrite, and though it usually looks white with soft-green streaks, it also forms with tinges of orange, lavender, yellow, pink and even black, according to Gemcyclopedia.com.
Nephrite
Nephrite resembles jadeite and is often used in its place in jewelry and tools. Generally, it is easier to find and more malleable for fine-tuned craftsmanship, though the final product is slightly more breakable than jadeite. Jadeite is actually a combination of the compound minerals actinolite and tremolite, which, according to the ICGA, usually is found in the middle- to dark-green hues, as well as in whites, reds and yellows.
Other "Jades"
What looks like jadeite or nephrite also could be one of several common variations of jade achieved by a process called metonymy. Some variations, according to Gemcyclopedia.com: bowenite as Korean jade; aventurine as Indian jade; californite as American jade; chrysoprase as Australian jade; or green amazonite as Amazon jade. Some of these variations mirror the look and structure of jade; others seek to be a unique variation.