In this book, Diamond never actually comes out and explains the difference between the Paleolithic and the Neolithic in so many words. He does not say "these two eras were different because..." Instead, much of the book is dedicated to understanding the implications of a change from Paleolithic ways (hunting and gathering) to Neolithic ways (farming).
The most important distinction between the Paleolithic and the Neolithic is that the Neolithic is the time during which agriculture was discovered and developed. Diamond describes the differences that this causes at great length in many parts of the book. Basically, he says that the discovery of agriculture led to more developed societies with higher population densities. This, in turn, led to some societies developing "guns, germs, and steel" while other societies did not.
The difference between these two eras, then, is that the Neolithic had agriculture while the Paleolithic did not. Diamond says much of later world history revolves around which areas experienced the change to the Neolithic first.