Using a similar application essay I used before
I honestly think there is nothing wrong in applying with a similar work of yours. And I think it, well, for all the reasons you've mentioned yourself, starting with the fact that it won't be considered as plagiarism, since the primary version of it is yours as well, ending with the fact that the work is sent to the same school.
However, depending on whether you've made only minor changes to it or not, you might adjust the work a bit more, or expand it, or formulate some thoughts differently for the sake of back-up (say, if there is a chance that any of the members of the admissions know or have previously seen your primary work - then they might think that you weren't generous on the effort and,therefore, don't perceive the opportunity to be worth working more on that work or on additional ideas). After all, you're the creator, so it is still up to you to decide upon all the abovementioned things!