The "burn test" can determine whether a fabric is all cotton or not. Basically you set a small piece on fire (be careful!) and see if the fabric turns to ashes (cotton) or beads up (polyester).
Cotton fabrics are preferred by quilters because they handle well. A cotton fabric can easily hold a creased edge for applique, for example, while a polyester fabric is much harder to crease. Polyester fabrics fray much more easily than cotton fabrics, so you want to be sure to provide quilting near seam lines to reinforce the seams and reduce stress on them from handling. Polyester fabrics are slipperier than cotton, so it's harder to get accurate points, etc. with polyester. Finally, polyester never fades and doesn't shrink, whereas cotton fabrics both fade and shrink a little. This means that the quilt does not age in the same way that antique quilts have. At the end of 20 years, the polyester fabrics will have their original color while all the cotton ones have faded togeter.