What have the Romans ever done for us?
The Romans are to blame for the name Palestine; they so renamed what was called Iudea after crushing Yet Another Jewish Revolt (TM). By that time the original Philistines were long gone. A classical example of digging up an old name from the history books to make a political statement.
The Arabs just took the name from the Romans when they conquered the area.
Essentially Palestine/Philistine comes from two different traditions of transliterating the same name into Greek. Palestine (with a π) is a secular Greek tradition whereas Philistine (with a φ) is the rendition by Greek-speaking Jews, which made it into the Septuagint (Greek translation of the Hebrew Bible); the Septuagint was in turn highly influential in other Bible translation including the King James. Hence "philistines" in English.
Note that what the Romans called Palestine was much larger than the original area of the Philistines, which was essentially just the Gaza strip.