Silk and cotton are both familiar natural fibers, but their structures are very different. Silk is an animal-derived fiber made mainly of protein, while cotton is a plant fiber composed mainly of cellulose.
Because of this difference, the same plant dye can produce different colors on silk and cotton. The result also depends on the dye, mordant, pH, temperature, pretreatment and other dyeing conditions. A Japanese study comparing the dyeing behavior of plant-derived dyes on several fibers found that protein fibers such as wool and silk showed high dye uptake under the conditions examined.
In this experiment at Hitotsuya(ひとつ屋), I dyed silk and cotton at the same time with the same plant dye and mordant to compare the results.
The materials were two similarly lightweight scarves. In the photograph below, the cotton scarf is on the left and the silk scarf is on the right.
Silk and Cotton Before Dyeing
▼ Cotton is on the left, and silk is on the right.

▼ A close-up of the cotton scarf

▼ A close-up of the silk scarf

The Comparison Dyeing Process
1. First, the scarves are scoured.

Scouring removes dirt, oils, starches and finishing agents that may remain on fabric or yarn before dyeing. For this experiment, I washed the scarves in heated water with a neutral detergent and then rinsed them thoroughly.
The appropriate temperature and scouring method depend on the fiber. Incomplete scouring can prevent the dye from entering evenly and may cause uneven color.
2. The scarves are dyed with beggar-ticks.

To compare the fibers themselves, I dyed the cotton without applying a soybean-protein or other deep-dyeing pretreatment. Both scarves were placed in the same dye bath at the same time.
In normal production, the pretreatment and dyeing conditions may be adjusted separately for cotton and silk according to the dye and the intended color.
3. The scarves are mordanted with copper.

Mordanting affects dye fixation and color development through interactions between plant colorants and metal ions. Different mordants can produce different colors from the same plant dye.
The basic mechanism is also introduced in How Natural Dyeing Works (Japanese page).
4. After washing and drying, the dyeing process is complete.

Results and Observations
In the photograph above, cotton is on the left and silk is on the right. Although they were dyed together in the same beggar-ticks dye bath, the finished colors are noticeably different.
The difference is related to the chemical structures of the fibers. Silk is a protein fiber, whereas cotton is composed mainly of cellulose. These materials interact differently with dyes and mordants, producing differences in dye uptake and color development.
Under the conditions used in this experiment, the silk became considerably darker than the cotton. This does not mean that silk will always dye more deeply. The result changes with the dye, mordant, pH, temperature, dyeing time and pretreatment.
A darker color also does not automatically mean better resistance to washing or light. Color fastness must be tested for each combination of fiber, dye and mordant.
The wool yarn shown on the right side of the photograph was dyed in the same bath. In this experiment, wool—a protein fiber like silk—developed a color similar to the silk scarf.
When I first began dyeing, I found this difference surprising. In everyday use, silk is generally considered more delicate and difficult to handle than cotton. In natural dyeing, however, silk can sometimes develop a deep color with comparatively little pretreatment.
Pretreating Cotton for Deeper Color
One traditional method used to deepen natural-dye colors on cotton is called gojiru, a soybean-protein pretreatment. Soybean protein is applied to the cotton to provide additional sites with which plant colorants can interact.
The traditional preparation involves soaking and crushing soybeans, straining the liquid, applying the diluted solution to the fabric and then drying it. Even application requires care, and some methods can easily produce uneven results.
A Japanese study of deep-dyeing pretreatments for cotton found that immersion in soybean liquid tended to produce uneven color, while brushing the liquid onto both sides of the fabric was more suitable.
Tannin is another possible pretreatment. A study from Ochanomizu University found that tannic-acid pretreatment increased the binding of an aluminum mordant and natural lac dye to cotton.
Cotton is therefore not incapable of taking natural dyes. It requires a dyeing process designed for the chemical properties of cellulose rather than the same process used for silk.