A way that helps people understand the difference:
Weave: how the fibers are assembled into a cloth.
Steel can be made into lots of things. A nice pocket knife and a car are both steel, but serve completely different purposes and are not interchangeable. Your cotton tee shirt and the cotton broadcloth you bought are the same fiber, but made into different products with different purposes. One is knitted, the other is plain weave. Same fiber, different construction.
Fiber Content: what the individual threads in any given fabric are made from.
Knives can be made out of a lot of different materials, steel, iron, bronze, silver, even bone or flint. Each one has it's pros and cons but they all achieve the same purpose, slicing stuff. Each one the end result is the same, although their appearance and ability to do the job at hand varies considerably, the task is still the same. You can buy jersey made from cotton, polyester, polyester-cotton blend, silk, linen, rayon or even wool but they all will have some stretch, a soft feel and a fine texture much like your tee shirt.
Clear as mud for ya now?