Well, there's more information in the works cited page than in the citation. You still have to do the cites with Chicago; they're just in the footnote and not cluttering up the writing. However, you can do one awesome thing in Chicago that you cannot do in MLA or APA, and that is the use of ibid. Ibid is so sweet. If the source you're citing is the same as your previous citation, you just write ibid. p. 34. No more writing, Cohen, Gutenweiller, Smith, et al. 5 times in a row.And just wait until you have to write annotated bibliographies. Those are so much fun. Ick.
I'm saying you can clearly see where I got my work from, from my citations alone. A works cited page provides more detail, but it's not necessary.