Is child psychology the same as developmental psychology?

Not really. Developmental psychology studies the development of the human mind. That is, how the psychology develops, changes, and comes to be. Also, in developmental psychology, usually, you study the development of the healthy mind in the belief that deviations from a normative (healthy) development gives rise to psychopathology (the unhealthy mind if you will). Child psychology is not really that interested in the normative. They focus mostly on the pathological child, and how to steer a pathological development in a more normative direction.

Another distinction is that developmental psychology is a branch of theoretical psychology, whereas child psychology is a branch of clinical psychology.