I'd agree, natural and organic aren't the same thing. They often have specific guidelines they follow to use various terms. Batting as others mentioned is a fluffy filler material, can be used for the stuffing in quilts, throw pillows and the like. No idea on that stuff in particular but the batting I've felt before was kind of funky feeling. It usually has backing layers on one or both sides (to prevent separation and fly away) and usually isn't all nicely aligned in one direction. Something you'd probably want for wicking.
Like the muji pads mentioned, the cotton is combed and all one direction. It either pulls tight and stops or feathers out. Batting is usually every which way with multiple layers. To each their own, I probably wouldn't use it personally but that's me.
This from a FAQ on a materials site that deals in bedding items (quilting, batting etc.)
Organic -
"Organic certification and true organic labeling can only be attained if the soil on the cotton farm (or where sheep are raised for organic wool) has undergone inspection by an accredited certification organization which has strict guidelines and standards. These farmers and ranchers must follow organic protocol that nurture and protect the animal and soil while prohibiting the use of chemicals such as insecticides, herbicides and fungicides that are toxic and harmful to humans and the environment. These toxic substances continue to outgas in fabric even after processing."
Natural -
"Natural or green labeling can be confusing and actually misleading. It sounds safe but the only requirement (for which there is currently no federal standard or inspection process) is that the end products are supposed to have not been treated with formaldehyde or bleached with toxic chlorine but there is no way to be sure. Cotton plants which are some of the highest sprayed plants in the U.S., can be treated with pesticides during the growing process and still have the end product labeled natural or green."
Referenced from this site
http://www.nofeathersplease.com/faqsandorderinfo
"Natural" isn't as wholesome as it leads people to believe.