The term "Deep Belief Network" has been abused in the literature (not pointing fingers, I've done it too). The DNNs used mean a neural net pre-trained with RBMs. Sometimes, when people say DBN, that is also what they mean. But really a DBN is a particular graphical model with undirected connections between the top two layers and directed connections everywhere else. The confusion comes from the pre-training procedure. The pre-training creates a DBN, which is then used to initialize the weights of a standard feedforward neural net. Then the DBN is discarded. It is a somewhat pedantic distinction. Since DBN is already an overloaded acronym (Dynamic Bayes Net) in the speech community and not entirely accurate for the pedantic reason I just mentioned, we decided to go with the DNN acronym.