What's the difference between deep belief network and deep convex network?
Deep convex network is relatively new architecture of deep neural networks, which has been developed in order to overcome the scalability limitations of deep belief networks (DBN). You can read more on technical details of the architecture and its performance in research papers, for example, in this paper as well as a relevant later paper, both from Microsoft Research.
It might be beneficial to read more about DBN on this fascinating peer-reviewed open-access site, as it's more comprehensive than Wikipedia on the topic. In my opinion, even more comprehensive and interesting overview of deep learning architectures in AI can be found in this technical report.