The most pertinent difference is frequency: 'via' is among the most common words in the language, whereas 'platea' is a Greek word that is rare enough that you need never use it and may never even encounter it.
'Via' is also a little broader, so to speak, in its sense, referring to anything from a mere path to a major thoroughfare, and is used in figurative senses much like our word 'way', whereas a 'platéa' seems to refer just to a broad city street.
Wondered about platea and curro, why there come no etymologically related word to my mind. Your comment explains the abandonment of platea for street (although it seem to have evolved into place (Platz)), but how did curro not survive in any form in English (and German). Italian, Spanish, French seem to all have a word for run based on curro. Jog, sprint, dash, race mean all in one way or another run but their roots seem not to be necessarily run unlike curro which comes from Proto-Indo-European *ḱers (to run). This was the word for run and the group of Germanic languages chose to forget it. A little strange to lose such an essential word even when considering hundreds of years of language evolution.