Carbon composition resistors have a rep for sweeter sound from their use in guitar amps. There is a grain of truth in that. Carbon comps have the largest voltage coefficient of resistance of any resistor - which, in English, means that as the voltage across them changes, so does the resistance, so they actually generate some distortion themselves. The distortion is pure second order, and hence sounds nice to guitarists.
The amount of resistance change per volt (and hence, the amount of distortion) is tiny. You need hundreds of volts of signal change before it gets audible. In tube-type amps, this can actually be audible under some conditions. Sometimes it's not even audible there, but folks who're looking for magic-mojo tone fixes love to sort through this stuff, like deciding their tone is better if they turn all their signal cords the same way or something.
On top of that, carbon comp resistors are not very uniform. They vary a lot from unit to unit in all kinds of ways, but also in the amount of the voltage coefficient of resistance. Some of them make a lot more of this distortion than others. That alone makes getting a magic-mojo resistor hard, even if the circuit lets it sound out.
In effects pedals with a typical 9V power supply, there just isn't enough signal voltage - or power supply voltage! - to make the real effect come through.
As to brown sound, tweed sound, and the other stuff, near as anyone can tell, that's purely an urban legend. It's true that the amps with the most brown sound, whatever people really mean when they say that, are usually very old ones. That means that they have the oldest, most value drifted, poorest consistency carbon comp resistors ever made in them.
Brown sound and/or other magic mojo in carbon comps, as related to low voltage effects, is purely advertising hype. There is no real reason, and no measurable effect to having them there, except for the excess noise they create. The placebo effect sometimes makes people do and say strange things, so it's sometimes happens that a person really believes that their effects sound better because there's a carbon comp resistor or two hidden inside their favorite pedal, but *that* chain of events is psychological, not electronic.
And that's the story.