I'm currently working on my PhD, and horn sharks are my study organism! I'm working primarily with horn sharks in the eastern Pacific (California Horn Shark - Heterodontus francisci, Mexican Horn Shark - Heterodontus mexicanus, and the Galapagos Horn Shark - Heterodontus quoyi).
There are taxonomic rules, guidelines, and even loose "regulations." The point of these rules is to keep some kind of taxonomic continuity...if people all refer to the same organism by five different names, a simple scientific conversation or correspondence can become a nightmare! The oldest valid names take priority; sometimes multiple people/researchers will identify and name a species over the years, and in that case the oldest name is given priority.
Sometimes it's necessary to completely reorganize or rearrange the taxonomic classification of an organism, like how megalodon has jumped around from one genus to another (Carcharodon --> Carcharocles --> Megaselachus, etc.).
Some taxonomic resources online will list most or all of the scientific names for an organism throughout the years. I think Leonard Compagno may do this for different shark species. This one is a good example:
Synonyms of Heterodontus francisci (Girard, 1855)
