I was in a book store and saw that there are two different paperback publications of "A Dance with Dragons". One published by Bantam, and one published by Harper. The Bantam edition is 1152 pages long, and the Harper is 704 pages long. The pages are the same size. What is the difference between the two versions?
Harper split A Dance With Dragons into two volumes for the UK market: you've linked to the first part, A Dance With Dragons: Dreams and Dust. The second part is A Dance With Dragons: After the Feast. Bantam published the entire book in a single volume.
Post-publication split into Dreams and Dust and After the Feast
A second form of split occurred involving A Dance with Dragons; while the book has been originally published as a single volume in the middle of 2011, starting around March of 2012 many English language editions were designed as two volumes (a similar decision to that taken in A Storm of Swords. The first half of these split editions are subtitled "Dreams and Dust", while the second half received the subtitle "After the Feast", supposedly because they include chapters that happen chronologically after the end of A Feast for Crows.
I have read both. There is little difference to warrant purchasing both. One or the other is sufficient. There is no more or less of the story of the "A Dance With Dragons" in either. What the After The Feast version has is a sampler of "The Winds of Winter" and a full appendix as you have seen in previous volumes. If you are collecting paperbacks, the Dreams version is the same physical book size as the previous volumes. The after the feast is in a paperback bantam sized book. The next volume in the series is "The Winds of Winter" and is expected to release sometime between February 2015 and 2017 according to Goodreads and Wikipedia.