Hi
regarding op-amp feedback circuits, the diode does two things: (i) forward conducts, and (ii) adds capacitance. I hadn't thought the capacitance was too important until now.
Most diodes have a fixed capacitance and a frequency-dependent component to their capacitance. Simple Si diode have a larger fixed capacitance and small frequency-dependent component. Schottkys and I *think* LEDs have more frequency-dependent capacitance.
A 1N4148 has about 5pF of capacitance, while a 1N4004 has about 15pF. In a tubescreamer, 15 pF is a significant addition to the 47/51pF feedback cap. At about 3 o'clock on the drive control (450k feedback), the standard cap gives rolloff at 7 kHz, but an extra 15pF lowers this to 5.4 kHz. That's half an octave difference. Many people would be able to hear that. If you put two diodes in parallel, it would probably tame the distortion quite a lot.
Schottky diodes (e.g. 2N5819) have low and frequency-dependent capacitance, because they don't have minority carrier injection and stored junction charge (the junction is a semiconductor to metal). Even more than the 1N4148, the capacitance is probably insignificant. LEDs have high capacitance at high frequency (e.g. 60pF at 1 MHz), buy I'm not sure low low it is at audio frequencies (e.g. 10 kHz).
cheers