What is the difference between symmetric bipartite graphs and a complete bipartite graph? I am studying Restricted Boltzmann Machines (RBMs), and it is described as a symmetrical bipartite graph. Link 

How is this different from a Complete bipartite graph? They seem to be the same to me, which is why I'm curious to why there is such a clear difference in terminology.
Your first link's usage of 'symmetrical bipartite graph' is indeed puzzling. From my knowledge of RBMs it would have been better for them to say 'complete bipartite graph' as you suggest.

The only formal reference I can find online for 'symmetry' in bipartite graphs is this link. But that does little to explain the usage in your case.