How does the Stockham FFT work?
Both algorithms are DIT, but the main difference is in memory access patern.
Cooley-Tukey - the most popular form of transformation from DFT to FFT comes from cache awareness, which is optimal for CPU pipeline.
Cooley-Tukey in first phase reverses bit order while Stockham changes order at each stage.
This is observable if you take look at edges - once you have half of the data it stays in place for Cooley-Tukey but moves for Stockham.

For CPU Stockham makes cache mispredictions while Cooley-Tukey makes thread serialization for GPU.

FFT - look at BFS vs DFS strategy.

FFT stage decomposition - very nice pdf showing butterfly explicitly for different FFT implementations.