The Tsukaito has two main goals: it serves as a grip and it holds the Tsuka tightly. A tightly wrapped Tsuka is extremely important for security.

Cotton allows better absorption of the sweat but it is less durable than silk and leather. It is the standard option for most Iaito.
Silk feels a little harder and is sometimes not recommended for beginners. However, it has a significantly longer lifespan.
Sweat is not really absorbed by Leather but this latter offers an excellent grip and it is the most durable material. Leather is available in standard and suede version (napped finish). The latter has a slightly better grip but it is slightly less durable.

Please note that those materials are made traditionally - they are not mass-produced. Therefore, colors can slightly vary depending on the batch.
In general, the Tsukaito is of the same color as the Sageo to keep the fitting harmonious, but this is not compulsory; you can select two different colors if you like.