The principal change in this new edition of Solow's well-known text is the addition of four appendices with "Examples of Proofs" from Discrete Mathematics, Linear Algebra, Modern Algebra, and Real Analysis. This probably reflects the feeling of many instructors that it makes little sense to teach students to do proofs out of the context of some specific part of mathematics. The appendices presumably allow instructors to choose one such topic and choose examples accordingly.
Solow's book attempts to teach students how to construct and understand proofs by creating a taxonomy of proof techniques and teaching them to students both by description and by example. Parts of this taxonomy are standard (e.g., "proof by contrapositive"), and parts seem to be Solow's own creation (e.g., "forward uniqueness").
The author stresses, and reviewers in the past have agreed, that this book will only be useful in the context of a course or seminar that emphasizes class participation. In particular, they claim that assigning it as supplementary reading would not be productive. I tend to agree, but also note that teaching from this book requires the instructor to "buy into" Solow's taxonomy and approach in a big way.
Solow's textbook is probably one of the best for instructors who want to approach the "Transition to Proofs" course emphasizing proof techniques rather than putting the focus on logic or undertaking a survey of advanced mathematics.
Fernando Q. Gouvêa teaches at Colby College in Waterville, ME.