Are the reasons for the differences in how men and women dress today the same today as they were in Ancient Greece?
In order to effectively answer this question, one has to make a clear distinction between the time of the Greeks and the modern setting. Recognizing that there is still a ways to go in achieving pure equality for women, there are more opportunities for women today than there were in Greek times. Being a woman in Greece was highly limiting. The contours of life for most women in the Greek setting involved a traditionally domestically bound existence and one in which rights were not conferred on women. The expectation was that women occupied a condition a bit higher on the social spectrum than slaves. For example, consider that in Sophocles' Antigone, part of what Creon cannot overcome in his mind is that a woman is questioning his edict in such a demonstrative manner. Antigone's own sister, Ismene, tries to get her to act more like a "traditional" Greek women. The manner of dress for a woman reflected this traditional and internally driven notion of being. The woman wore a tunic that covered most of her body, and a shawl that was worn over the head. Essentially, the style of dress sought to limit her interaction with the outside world. Her domain was not to be in the outside setting, but rather remain at home, in the realm of the internal and domestic. This inward condition is conveyed by the clothing and dress of the day.
I am not sure that the exact same conditions of servitude exist in modern dress today. Modern fashion does not reaffirm a political condition of being. Men and women are seen as consumers with purchasing power. Fashion is marketed to them. The difference in the use of revealing clothing can be seen as an element in which there is a difference in style and fashion. There can be many different points to make on this, such as why sex and revealing clothing is a part of women's fashion and whether or not this is the same standard in men's fashion. However, there is not a political reaffirmation of gender roles seen in modern fashion as it was in Greek use of clothing fashion. This would constitute as one reason as the differences in how men and women dress today as opposed to what was seen in Greek times.