In this question: Should the [deep-network] tag be replaced by [deep-learning]?, it seems that it was agreed upon that the deep-network should be replaced by deep-learning.
I think that "deep learning" is already a standard expression, so this tag should be the main tag. The tag deep-network could also exist on this website, given that the expression "deep neural network" is also common, but I would not say that it is a synonym for deep-learning. A deep network is a network that is deep (that is, it possesses "many" layers). However, deep learning is not just concerned with the architecture of the NNs, but it also concerned with the learning part.