I went to a wedding last Thursday up in Monsey and during the meal, a Rabbi got up to speak. His speech was about why a bride wears a veil at a wedding. His main point was that Jews, unlike non-Jews, don't have relationships before they are married, meaning that the man and woman don't think of themselves as a unit until the end of the חופה. Therefore, she wears a veil to show that they are separated until the ceremony ends. With all due respect, you could ask yourself why a woman covers her face and not a man, and while one could provide many answers, like the woman being in a veil is a sign of modesty, it is clear that wearing a veil is another remnant from the ancient world. At most weddings in most cultures, there is some sort of veiling process. In Judaism, we make it into a whole ceremony, which we call the בדקין.
Furthermore, what struck me as really interesting was when the rabbi tried to say that a veil was a Jewish specific thing. Even if you were to ignore other historical accounts and focus yourself specifically on the biblical accounts of marriage, it is pretty clear that a veil is not a "Jewish" thing. In fact, the first description of a marriage, between Jacob and Rachel (which really became Jacob and Leah), one could say the switch occurred as a result of Jacob not seeing his bride's face before the ceremony, thus allowing Laban to pull the switch.