This depends a lot on the make, model and year of the washing machine. The different programs changes the sequence of the machine, so how long it lets the clothes soak, how hard it works the clothes, how hot the water is, etc. And this is tested and optimised by the manufacturer for the different types of clothing for each model before it is sold. In general the main difference between whites and colors is the temperature of the water. Some pigments might not handle very hot water and release from the fabric, even attach to another fabric. Traditionally the biggest issue have been with red pigments. White clothes do not have pigments and can therefore be washed at a much higher temperature. This might also reduce the washing time as the dirt dissolves faster in warmer water.
The settings mostly affect water temperature, cycle time, and spin speed. Water temperature is the primary difference between cycles.
Colder water helps reduce dye bleeding, but also tend to clean less. These effects are drastically reduced with modern dyes and detergents, with dyes bleeding less and detergents cleaning better with colder water.
Slower spin results in soggier clothes but also cause less damage to the fabric and fasteners. This is normally countered with increased spin time or more drying in later stages.
The exact details will vary widely by machine.
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