Polyurethane laminate (PUL) is a polyester interlock knit fabric that has a very thin film of polyurethane laminated on one side.  It is useful as a water barrier but is very thin and breathable and can be machine washed and dried.  It doesn't feel plasticky like PVC; it's very soft.  Cloth nappy outers are often made from PUL.

The absorbent pants have a layer between the outer fabric and the towelling gusset so you cant feel it at all.  The baby mat has a PUL backing and edging, but it feels extremely soft.