Hand woven table linens include placemats and runners in linen, cotton, and, for heavy patterns, wool. The main difference between the two is that runners are narrow and long, compared with placemats.
Towels are linen and can be utilitarian or fancy tea towels. Fancy tea towels are an invention of the 19th century. They are useful and attractive used to wrap warm bread or line a tray. The utilitarian towels in small patterns are drawn from my study of 17th c. towels for the "ordinary" home. They are in different weaves from the diaper family, small all-over patterns. And, no, the name of the cloth came first, and then the article for babies!
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