If I am going on, say, a website such as this? I am perfectly comfortable saying who I really am, were anyone to want to know. But at the same time, I feel that it is not in my best interest to put it out in the open as a default - kind of a "ask and I will tell, otherwise we will leave it at a nickname".
So I suppose, by default, I set to anonymity. Not putting real info out in the open online.
Why? Comfort and security. Simple as that. If I am relatively anonymous, then I have no problem with harassment, stalking, generally creepy-ness, etc. [which I have had some issue with IRL when not being anonymous].
But, don't take that as a general answer for my opinion for all people. I feel that both have a place, both have advantages and disadvantages. In some case I feel it best to go with openness, in others to default to anonymity.
[Yes, I am being purposefully indecisive. Sorry.]