It's not clearly explained why Twilight Sparkle and her friends are the only ones who can use the elements of harmony. Twilight tries to ask the Princess why she can't use the elements anymore and she's like "uh, I just can't".
I just assumed that it's because the elements of harmony require a group of best friends. And all of the Princess' besties died from old age years ago. And now she's too famous and powerful and has too many responsibilities to ever make friends again. Not besties anyway.