I would start by reviewing each story one at a time. Identify the theme first. (The enotes study guides for each story will help there; see the links below.)
Then start identifying the elements of style in each one. For example, look at the sort of language used in "The Lottery." Is it simple or complex? Do men and women speak the same way? What sort of metaphors or images are used?
Then go through the Faulkner story doing the same thing. Then, once you've looked at the key components of style in each, draw conclusions. One uses more _ (fill in the blank) while the other is more (fill in the second blank).