The difference being there is a fish called The Australian Rainbow, but as far as I know there isn't a fish called The South American Tetra
There is no fish called the Australian Rainbow. This is not a scientific species name, so it is totally irrelevant. So is the name "neon tetra" for that matter. It just so happens that enough people "assume" neon tetra refers to Paracheirodon innesi, but that does not make it any sort of formal name. In North America, "cardinal tetra" is generally a common name for P. axelrodi, but in Europe this fish is (or used to be) known as the red neon, not the cardinal. The similar species P. simulans is called false neon and green neon. More than once I have seen hobbyists get these names mixed up with "green fire tetra" which seems more often to be used for a totally different fish, Aphyocharax rathbuni, but they thought it was P. simulans.