According to my understanding for polynomials of degree lower than 5 the solution should always be expressed in terms of radicals. My version of Mathematica is 10.0.2.0.
You can use ToRadicals to convert it to the traditional notation with radicals:
ToRadicals@Root[1 - #1^2 + #1^3 &, 1]
Yes. You can write the root of any polynomial of 5 or lower with radicals. But very often you don't want to.
Thank you Sean, I was not aware of function ToRadicals. This solves my problem.