The definitions for nursery and playroom and the context of the story make it clear why Bradbury used the term instead of calling it a playroom. The Oxford English dictionary defines a nursery as "a place where young children and babies are taken care of while their parents are at work" and as "a room in a house where small children sleep and play." On the other hand, a playroom is simply described as "a room intended for children to play in."
A nursery, regarding the definitions, better defines the purpose of the room the Hadleys had built for their children. Since they are busy parents, the room serves as a substitute for parental care. It is, therefore, not simply a place for the children to play in. The expectation most certainly seems to be that it will also provide some form of nurture and care and give the parents more time for themselves.
It becomes apparent that the children's obsessive dependence on the nursery alienates them from their parents. The room provides them with everything they want. Not only that, but they can instruct it and exercise their every whim. The nursery is so unlike their parents, who they have come to resent for meddling in their fun and not giving them what they demand. The room has become, to the two spoilt children at least, a better caregiver than their parents, as David McClean, the psychologist, tells George:
This room is their mother and father, far more important in their lives than their real parents.
In this context, then, nursery is the more fitting word to describe the room.