Does unpublished paper matter for U.S. graduate school application?
How different are papers that are "in draft", "in review", and
  "pending publication" for graduate school application?


They all have value, although it is hard to quantify. In draft papers show you have initiative and signal (but not reliably) some level of creativity and experience. In review papers suggests the same things to a stronger extent as it implies that the work is more mature (a draft could be at any stage of completion). Obviously, a paper pending publication is a very strong signal of ability.


  Or alternatively, I could submit my draft and wait for the reviewer's
  opinion, so I can put mark the paper as "in review". This will serve
  as a proof that my work is mature enough to be seen by others. But is
  it worth it?


Without knowing much about the details, I think it would be worth doing if you are not going to lose out in someway. I know of people who got academic positions partially on the basis of papers in review (although usually at advance stages, so second or third round). In contrast, I don't know of anyone gaining much advantage from claiming they had a draft paper.