Typically exponential functions, being e^x and ln(x), are introduced later in the text after differentiation and integration are already exposed and discussed in detail. This is because the exponential and logarithmic functions can be more easily defined with the use of those techniques.
An "Early Transcendentals" book, covers these functions from the beginning and uses them, and assigne exercises with them from the get go, even when differentiation and integration are being taught and used for the first time. The reason for this book is if a certain curriculum is designed this may. For example, maybe engineers or physicists need to know about these functions right away, so may they should take a course that covers these as soon as possible.
It only makes a difference in the earlier, say the first seven or so chapters, in a book with likely around sixteen chapters, so it makes little different in the big scheme of things, especially once you get to multivariable calculus.
I would sugest that you make sure the book is the full version (the list price should be around $135), and not just the first volume, otherwise you won't get the multivarible stuff in the same book.
I used the normal version of Stewart and it was fine if you don't need to know much about the theoretical background (but even Stewart dips a bit into proofs and such).
I would suggest you get the normal version too (especially if you don't really understand the difference) if you're teaching yourself, as there's no reason getting the less commonly used verison, as most courses that I know of are not structured with an early transcendentals syllabus.