It seems to me that children can be such vicious little creatures because they don't have any sense of self-worth yet. So they try to gain it by tearing down or controlling everyone around them. They can also become parts of cliques for the security and confidence that being in a group gives. The friction at these groups' boundaries tear outsiders down while protecting the group's members. This gives an impression of superiority and feeds the members’ needs for self-worth.

So at a basic level, the cruelty exists because the children do not yet have anything to be proud of. They take pride in enforced superiority, created by destroying the dreams of those around them.

Of course, this cycle never even begins if the children in question have something to be proud of, something that is really and distinctly theirs. But the young rarely have this level of differentiation.

Note: By cruel or vicious here, I don’t mean completely blunt or anything of that ilk. There are times when children act honestly sadistic in ways that put most adults to shame.

Clarification: I realized (from looking at wrinkly's writeup) that I should mention I mean elementary school-aged individuals whenever I say child, sorry for any confusion.