The question of why to study ancient philosophy seems to me to be the same as the question of why to study philosophy at all. Philosophy may be uniquely suited to permanent relavance from one age to another as the central issues to philosophy are largely intractable, perenial, and basically unchanging.
We study ancient philosophy for the same reason we study contemporary philosophy - to inquire into and consider issues of what it means to be human, the nature of morality, and other conceptual notions regarding the human condition.
The signficance is getting to know what issues have been set forth and discussed so far. Reading the Ancients allows us to access several things:
1. It gives us insight into a world view that no longer exists in many ways. Getting under the skin of Aristotle and Plato allows us access to seeing the problems of philosophy in a way most moderns do not.
2. It allows us to keep track of the work that has been done on a philosophical problem. Whether the issue is how to act or what consititutes knowledge, the ancients most likely spent considerable time on it. From this work, we can follow the issue through history and into the modern arena.
3. It provides an opportunity to appreciate the "human condition". Many issues we face today have been around for a long time. What does this fact tell us?
Studying ancient moral philosophy (like Aristotle's ethical writings) or political philosophy (Plato's Republic) gives us a window on to the past. It tells us what people from ancient times valued. It can also give us an outsider's perspective on our own society. By this, I mean that we can use these ancient thinkers' works to evaluate our society from a different perspective. When we only think of things from our own perspective, we get narrow. By reading and using ancient ways of thinking, we can broaden our perspectives.
this is probably one of the issues tha people do not regard, that ancient philosophy is very much alive in the day to day life of the present world
A colleague of mine once stated that all philosophy after Plato is a footnote. Ancient philosophy is not so ancient. There is a crucial "present" in much of these writings.