Why do we continue to use names from these ancient languages?

It dates back to Aristotle (384-322 BC) and Linnaeus (1707-1778), when early biologists were trying 1) to understand the relationship between living organisms and 2) to name them in a way that expresses that relationship.  It was a way of putting species into categories that made sense.  Plants also have common names but unlike, scientific names, common names are not always specific.  One plant may have multiple common names or two plants can share a common name.  

By intent, one plant should only have one binomial scientific name.  However, every now and then someone discovers that more than one person reports a 'new' plant and names it.  Also, names can evolve as we learn more about plant relationships through genetics and other insights.  Therefore, we need authorities to help us keep this all straight.  One such authority is the Kew Royal Botanical Garden, and they treat this issue in an open academic way by referencing what they believe to be the original report of a plant and they list all of the names that are considered synonyms as a result.  See their report on Osmunda claytoniana.

That said, there are other authorities.  I volunteer at two local universities, and they don't always agree on the official name.  (NC State uses POWO and UNC-CH uses FloraQuest.) There still should be no confusion though because the accepted name and synonyms still identify a unique plant (almost all of the time).