Pyro vs Edward: like Edward, Pyro is a deep probabilistic programming language that focuses on variational inference but supports general composable inference algorithms. Pyro aims to be more dynamic (by using PyTorch) and universal (allowing recursion). In particular it should be easy to integrate existing simulator libraries into Pyro as long as they have a Python interface. The Pyro, Edward, and Probabilistic Torch teams are collaborating to identify common concepts and to ensure models are maximally compatible across our systems.
Pyro vs PyMC, Stan: Pyro embraces deep neural nets and currently focuses on variational inference. Pyro doesn't do MCMC yet (but we're actively working on HMC). Whereas Stan models are written in the Stan language, Pyro models are just python programs with pyro.sample() statements. Stan caters to the stats community who value interpretability and model criticism. Pyro caters more to the ML community, and tries to add some statistical tools to existing PyTorch models.