This is not a textbook on celestial mechanics, and it makes no pretense of being one. Although there is a good deal of explanatory information and diagrams, particularly in the discussions of orbits, the authors state right in the preface that the calculations are "recipes," and that they mostly consider only the essential factors and ignore many of the more complicated corrections needed for the most precise results. In spite of this, the results of these calculations are typically accurate to within about a tenth of a degree or better, which is well within the field of view of a telescope at medium powers.
The main change from the previous edition, as the title indicates (and the authors explain in the preface), is the addition of spreadsheets for nearly every calculation. The spreadsheets themselves can be downloaded at the publisher's web site [...], and include macros for high precision computations as well. For readers who are not familiar with using spreadsheet programs such as Microsoft Excel and the freely distributed OpenOffice Calc, the book includes an introduction showing the basics of how to use them.
For the most part, the math is limited to high school trigonometry, e.g. sines, cosines, and tangents. The one exception is the section on coordinate transformations using matrices, but this section supplements other methods and is accompanied by a brief but sufficient overview of what a matrix is and how to multiply matrices for those who are not familiar with them.
While I can certainly think of additions I would have liked to see in this new edition (such as discussions of asteroids and trans-Neptunian objects, lunar librations, and an updated comet table), all in all, this book does a good job at what it sets out to do. The formulas and algorithms are presented clearly and with an appropriate level of detail, and the authors walk you through all of the examples in step-by-step detail, showing you the intermediate results you should obtain at each stage. The spreadsheets do a good job supplementing the text and demonstrate an additional tool for performing the calculations.