Understandably RNNs are very good at solving problems involving audio, video and text processing due to the arbitrary input's length of this sort of data.
What I don't understand is why RNNs are also superior at predicting time series data and why we use them over simple MLP DNNs.
Say I wanted to predict what the value in the time series is at $t+1$. I would take a window of, let's say, $t-50, t-49, \dots, t$, and then feed loads of sampled training data into a network. I could either choose to have a single LSTM unit remembering the entire window and basing the predictions on that, or I could simply make a 50 neuron wide MLP network.
What exactly is it about RNNs that makes them better in this scenario or any scenario for that matter?
I understand that the LSTM would have substantially fewer weights (in this scenario) and should be less computationally intensive, but, apart from that, I don't see any difference in these two methods.