For me, paper piecing is when there is a pattern printed on the foundation. You can use paper, fabric (iron fabric to freezer paper and run it through your ink jet printer) or use non-woven sheets (June Taylor has a product that you can use for this) that you can run through an ink jet or lazer printer. If you use paper, you would remove the paper after all the blocks have been sewn together. If you use fabric or the non-woven sheets, you would leave them in.
Foundation piecing is when you are sewing fabrics together without a pattern, as for string quilts, coin quilts or crazy quilts. The foundation can be fabric or paper, or non-woven sheets. Some people also use pressed (flattened) coffee filters or used fabric softener dryer sheets. If you use paper, you should remove it, but fabric or non-woven materials can be left in. I have talked to quilters who use adding machine paper, tissue paper, TV guide pages, yellow pages, paper like you find in a dentist or doctor's office on a roll or just plain photocopy paper cut to size.
Of course, as was said by someone else, all paper piecing is foundation piecing, but not all foundation piecing is paper piecing.