See David P. Roberts' review of the first edition (2000). Besides the move from Birkhäuser to Springer's Undergraduate Texts in Mathematics series, the main change for this edition is that the original chapter 8, which started from the Peano axioms and included constructions of the integers and the rationals, has been removed. The author says that this "was always somewhat out of place given the level and scope of this text." This material has moved to Bloch's (forthcoming?) The Real Numbers and Real Analysis (Springer 2011).
Several other changes have been made:
Preface to the Second Edition Preface to the First Edition To the Student To the Instructor Part I. Proofs 1. Informal Logic 2. Strategies for Proofs Part II. Fundamentals 3. Sets 4. Functions 5. Relations 6. Finite and Infinite Sets Part III. Extras 7. Selected Topics 8. Explorations Appendix: Properties of Numbers Bibliography Index