or is it called a neural network because it is "fancy"?
Machine neural networks are called such because they deliberately emulate the functioning of biological neural networks, in an attempt to find ways to solve problems that are not easily solvable by "traditional" signal or data processing means.
Is there a difference between a linear/nonlinear adaptive filter (chain) that approximates an unknown system and a "neural network" that does the same
There is certainly a difference in structure. A feature of both biological and machine neural networks is that they take a large number of inputs (different from a typical "filter"), convolve those inputs with some weighted taps (same as a typical "filter"), then run that result through a nonlinearity to get a "neuron fired, neuron not fired" result (way different from a typical filter).
If you wanted to implement a linear filter with machine neural networks, you'd typically end up with way more "neurons" than your filter would have taps. In biological systems that do this sort of thing (like the more-or-less wavelet processing that happens in your inner ears, before the information gets to the big squishy biological neural network between your ears) "linear filtering" is carried out by either specialized neurons, or other systems (mechanical in case of your ears, electrochemical in case of your eyes, etc.)
Is a neural network an adaptive filter?
That depends on the neural network. A machine neural network, by itself, does not adapt to changing stimuli. The typical machine neural network, as deployed, does not change the weightings or the activation functions -- those are determined during a training phase. Making a neural network that can learn involves adding specific features to the system, and training a neural network is a process that often has unexpected results, so it's not generally something that is allowed to happen unmanaged in the field.