White noise carries every frequency at equal intensity, which makes it feel a bit harsh and sharp. Pink noise softens the higher frequencies slightly, giving it a gentler, more natural texture closer to rain.
Brown noise leans into the lower frequencies, producing a deeper, fuller rumble, closer to a distant waterfall or wind.
For focus, most people find pink noise more comfortable. For falling asleep, brown noise's deeper texture tends to be preferred. White noise is most effective specifically when you need to mask an already noisy environment.