The major difference between this edition and the preceding one are the following: There is a better discussion of assigning probabilities to events, including introducing independent events and Bayes' theorem in the text. The consideration of random variables and their expectations is greatly improved.
I own both the 5th and 8th editions, and each has their own merits.
I had used the 5th edition when I was first studying the material and find myself returning to it again and again because of its excellent exposition. The authors do a great job of walking the reader through the material, developing the ideas gradually and helping you discover the theory alongside them. It places real emphasis on intuition and understanding, rather than just formalism, which makes it an incredibly enjoyable read. It is easily one of the best math textbooks I have come across, making it a personal favorite.
The 8th edition, on the other hand, I mostly use as a reference book. It is very comprehensive and much more detailed, covering not only the classical foundations of mathematical statistics but also more modern topics. The notation and overall mathematical presentation are much more formal, a noticeable shift away from exposition the earlier edition. Because of this, i think it is best suited for readers who are already comfortable with the subject and know what they are working with, or for the more technical and formally inclined reader.
Very clear and easy to read. Really great if you are transitioning from pure math to industry and are looking for something that is easy and quick to read.