I am still at the beginning (just bought the book, and my focus remains on preparing a classical recital for now) but the first lesson is about the various jazz rhythms, and how they are responsible for the "jazzy" sound. The author also stresses that jazz is about change (little predictability) and how you can create a dialogue between two melodies. Again, "jazzy" sound here. I have not reached the chords part yet.
This book is supposed to bring you to grade 5 of the ABRSM jazz program, to give you an idea - but then again it depends on discipline and motivation (my previous attempts, with other books, have not been glorious). The book emphasizes group practice: most exercises can be done either alone or, preferably, with a friend. I imagine that you should be able to play something decent after a couple chapters.
In my vague memories, Jazzology is a very dense book that could not do it for a self-learner like me. I do not know the other book, though.
Regarding your other question, I do not plan to buy another book - I would already be very happy to finish this one seriously - and if needed, the Levine book is there. The next step would be to take actual lessons with a jazz specialist.