Maybe it's best illustrated by these two pictures (which sometimes say more than 1000 words):

[![enter image description here][1]][1]

[![enter image description here][2]][2]

The first picture represents stress, of which there are three different kinds, as you can see (therefore stress is a second order tensor quantity).

The second picture represents traction. The red $\tau$ vector is the component of the traction in the direction of the surface. 

As you can read, in the definition of traction your book is talking about a force **on** a surface, while in the definition of stress force **across** a surface is mentioned.


  [1]: https://i.sstatic.net/5L6bj.png
  [2]: https://i.sstatic.net/7wR7K.jpg