Computational resources is always the bottleneck in bioinformatics, quantum chemistry, or any sort of high data volume analysis or simulation, and the FFT is a fundamental and commonly used transform in all fields.

At least for people who still use computers to, well, compute things, a faster FFT is a huge deal.

I'm not sure what you mean by "on a small window such that this isn't really applicable"; can you give an example? As long as you accept the inherent time-frequency resolution trade-offs, there's no obstacle to using FFT on a small window. It's called the short-time Fourier transform (STFT), and it's used everywhere; it's probably used more than analyzing an entire signal, since we frequently want to know how power and phase change over time in a signal, and a full-signal FT can't tell you that.

FFT is very useful on a small window, but algorithms that improve the asymptotic efficiency are unlikely to be useful at that scale. With n=100 the asymptote doesn't matter. We'll still be using the FFT forever, I'm just skeptical that the frontiers of technology will be advanced by a faster FFT, since it seems like the coolest stuff is happening elsewhere. It's increasingly becoming the quick-and-dirty counterpart to the sophisticated-but-slow methods.

This is a useless post except that I wanted to highlight that wonderful little bon mot. Well done.