I was intrigued when I heard about the 30-year old race up the Empire State. There is apparently a whole class of stair running professionals and several "run-up" races, including the Willis Tower in Chicago, CN Tower in Toronto, and the Sydney Tower.
One veteran Empire State participant told me, "they are really fun in a sort of sick way."
The coolest part is reaching the top, often an observatory deck, and looking down to grasp the vertical climb you accomplished.
As a workout in your training program, stair climbing is one of the best ways to improve your VO2 max and strengthen oft-neglected lower leg muscles. Especially if you don't have easy access to hills, it mimics uphill running with less impact, but also less ventilation! If done at lower intensity for a longer duration, it can be used to effectively increase your aerobic capacity.
But descending can be more stressful on the legs, so slow down and use it as recovery ;).