At a glance it's very hard to distinguish the two as the veils look similar (the more "popular" niqab you see everyday in the media is just one variation of hijab).
So what is with this similarity? Has Syriac Christians been wearing veil since the religion spread in Eastern Mediterrania? If they were, was it a culture unique to Christianity at that time or was it adopted from existing culture - Roman or Persian maybe?
Edit: I'm speaking of Syrian Christians in Middle East of course. I don't know much about the situation in USA/Europe.
I don't have sources at the moment - I'd have to trek off to a library, but at least some historians have hypothesized that the proximate origin of Muslim Hijab is in Byzantine custom. To one degree or another, women in Greek (classical, hellenistic and byzantine) society were segregated and at times veiled. The same is true of ancient Persia. A glance at the costume of women in late antiquity and medieval europe shows that the way they dressed was not unlike a fairly conservative type of hijab and is reflected in the dress of Christian religious sisters (nuns). However, at this point it's not easy to say whether in fact the tradition of wearing a head covering in Syriac society is due to cultural influence from Islam or whether it represents simply the continuation of ancient tradition.
In a sense, the more general point is that purdah/hijab preexists Islam but that it has definintely been reinforced in some circumstances by Islam.
In Turkey, Syriac Christians only wear their (thin, gauzy) veils in prayer in obedience to 1 Corinithians 11:
4Every man who prays or prophesies with his head covered dishonors his head. 5But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved. 6For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head. [...]
13Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered? 14Does not the very nature of things teach you that if a man has long hair, it is a disgrace to him, 15but that if a woman has long hair, it is her glory? For long hair is given to her as a covering. 16If anyone wants to be contentious about this, we have no other practice—nor do the churches of God.
When I went to Syriac services in Mardin and Midyat (the two main towns in Turkey with Syriac populations), the women seemed to wear their covering in prayer only because they immediately took it off as soon as they stepped out of church. The style (specifically, the fabrics) and exact way they wore their headscarves (with ample hair protruding from the front) meant there was no way you'd mistake them for the Muslims (mostly Kurds and Arabs, with relatively few Turks) who made up the majority of these towns. This sort of hair covering is common in many forms of Christianity, though not in most forms that descend from Latin Chrisianity like Catholicism and the various Protestant groups, and I don't know the history of Latin Christianity and hair covering. Many women in Eastern Orthodox societies do cover their hair in prayer, even ones that never experienced Muslim rule. It's fascinating to watch documentaries about Russian oligarchs who have built their own private churches, and while the documentary crew is documenting the man's piety, his fashion model wife is by his side, head-covered. She would be uncovered for the previous and next segments, and only cover her hair in the church.
I've seen ideas that the early Christian practice of women covering their hair in prayer comes from a Roman practice, which seems to me likely, though it's far from my field. This may interest you. The Talmud also records that a woman going about with her hair uncovered (the text might also mean "untied", but has traditionally been understood as uncovered) is legitimate grounds for her husband to divorce her.