With the staircase, you can only go from one floor to the floor above or below it; with the elevator, you can go to any floor of your choice with a single click. So instead of going from the basement to the first floor, to the second floor, to the third floor, you can just zip from the basement to the third floor. Upgrading the staircase to an elevator costs 250 Diamonds.
NOTE: Buying the elevator does NOT buy you an additional floor in your store.