ANN vs RNN for time-series prediction? The basic argument against ANN for time-series is that there is no conception of time for the inputs. However, it is obvious that one can feed multiple time-steps into an ANN by using various lag lengths where the features are just various lagged series. And "long term" memory one could include the feature of a moving average.
I understand RNN's have ability to work with variable length input. But given the case of an ANN with lookback lags and a fixed-window, what is the advantage of the RNN over the ANN?

The biggest practical difference from the prediction point of view is that ANN would use a fixed number of timepoints while RNN theoretically has no limit on how far it can look into history. Additionally, if you use ANN, the number of parameters would grow with the number of time-points you consider, while for RNN it would not. What follows, ANN would need more data, more computational resources, and would be more prone to overfit.
But if you look at the results of forecasting competitions you will notice that traditional or hybrid forecasting methods usually outperform neural networks, so in many cases, the answer would be that you should use neither.
