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.