What is the difference between Slow Feature Analysis (SFA) and a Moving Average? I have started to read more about Slow Feature Analysis and I was wondering how SFA differed from simply taking a moving average?

The linked article suggests, "SFA is an unsupervised algorithm that learns (nonlinear) functions that extract slowly-varying signals from their input data.".  

I understand that it is certainly more complicated than just taking a moving average on a signal, but I don't really understand what the benefit or purpose is of the learned non-linear functions.

What would be some typical applications of SFA?
Mathematically, SFA differs from moving average by the fact that the instant output $y_t$ can only be a function of instant input $x_t$. On the other hand, the output of moving $y_t$ is a function of the past history of $x_{t'}$ with $t' \leq t$.

One immediate result due to this difference, is that if you have a flat constant input, with only a pulse at time $t$, then the pulse will be smoothed if you use a moving average, but its shape will be kept better if you use SFA. 

You can always try both moving average and SFA for your task, although the moving average requires some input parameters such as window width while SFA does not.