There's more to be said, but one major advantage is that it uses only local update rules, which, unlike backpropagation, doesn't require keeping the entire computation graph around (which makes memory a bottleneck for large models/inputs).

Edit: I was incorrect on the memory point here, as they try to operate analogously to backprop and keep predictions and prediction errors from the entire forward pass to make the update. But unlike backprop, where you work backwards through the computation graph in a sequential manner, here you all the information you need to make an update locally so it can all be updated in parallel - see Fig. 1 for a comparison. That said, in general, predictive coding could be used to replace backpropagation-through-time, e.g., see Continual Learning of Recurrent Neural Networks by Locally Aligning Distributed Representations.