The difference mostly lies in how they are trained. DBN are pretrained to reconstruct the input and then fine-tuned with back-prop, while DNN has a pure supervised training with back-prop. However, I think one could say that DBNs are a subset of DNNs. DNN is a general concept which refers to a neural network with a lot of layers. The pretraining of DBN has shown to be beneficial if the training set is small. 

http://stats.stackexchange.com/questions/51273/what-is-the-difference-between-a-neural-network-and-a-deep-belief-network

The difference mostly lies in how they are trained. DBN are pretrained to reconstruct the input and then fine-tuned with back-prop, while DNN has a pure supervised training with back-prop. However, I think one could say that DBNs are a subset of DNNs. DNN is a general concept which refers to a neural network with a lot of layers. The pretraining of DBN has shown to be beneficial if the training set is small. 

http://stats.stackexchange.com/questions/51273/what-is-the-difference-between-a-neural-network-and-a-deep-belief-network

Nice reply from Erik. To add further, DBN is formed of stacked Restricted Boltzmann Machines (RBM). Each RBM is a two-level model, a layer with visible units and a layer with hidden units. 

DNN is a totally different concept which refers to neural network with multiple number of hidden layers.

The difference also lies in the fact that the RBMs are unsupervised models while neural networks are not.

Another aspect to consider is the use of Convolutional Neural Networks for deep learning. They contain convolutional layers connected to a fully connected MLP, these convolutional layers form the 'deep' aspect. This can be further seen in work of Mnih (2013) which uses a CNN for deep reinforcement learning.

Also see: http://www.deeplearning.net/tutorial/

Thank you to everyone,  Mr Erick  i followed the link, can i say that DBN is a type of DNN?

Yes. DBN is a one type of DNN.

For what kind of data which model would fit?

Deep Belief Network is constructed through the stacking of RBM and RBM is a generative model which has two layer such as visible layer and hidden layers while DNN is a kind of neural network which has more than one hidden layer in combination with input and output layer.