The advantages of an elevator are that it can be powered by a renewable energy source and there are no rocket casings falling back to earth. Any fuel that you take up into space on an elevator can be used outside the effects of atmospheric conditions where it is most effective. More to the point, you can build up a stockpile of the stuff in space without burning rocket fuel to get it there. When you come down the lift you don't have to worry about the heat of re-entry because the speed of descent is limited.
There are enormous challenges in building such a structure but if we want greater lift capacity all the time (which means it becomes increasingly cheaper to send things into space or bring them back) then it's the way to go.
