Like Jasmin says, it's just a question of formality. 'children' is formal. 'kids' is informal.

There's not really a difference but child is more formal than kid

I think "child" also can have a connotation of being younger or less mature than a "kid", depending on the context. Teenagers will be insulted if you call them children :). "High school children" sounds very weird. But saying "high school kids" is very common (ages 14-18 or so).