I've only worn a veil very briefly, when I got married. I did it for cultural reasons, which mainly involved pleasing my mother.
However, I have worn a mask. At first I felt very self-conscious, but then I found it very liberating when I realised that (apart from people who knew me and were used to masks, and who would know me by the way I stood and moved) no-one would recognise me or be able to identify me later.
Maybe people wearing veils feel the same.