Q:

Why not write the solutions of a cubic this way?

For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}.$$
but this is ambiguous because it does not tell you which cube roots to match up. Why don't people write it this way today:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\frac{-p}{3\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}}$$
which is unambiguous.

A:

Since no one has posted an answer and since my comment is a sort of tangential answer (and relevant to another question): 
In UCSMP Precalculus and Discrete Mathematics, 3rd edition, p553 (in the "exploration" question), the cubic formula is given in a form analogous to what you describe (though it is for the general monic cubic, not the depressed cubic). It is given that way for the reason you describe—in particular, because of the way that most calculators and computer algebra systems define the principal root, the "traditional" way of writing the formula does not always yield correct results when computing blindly with technology.  The formula as printed in the first printing run of PDM is actually missing a term, though it should be correct in subsequent printing runs.  The correct formula reads:
Let $$A=\frac{\sqrt[3]{-2p^3+9pq-27r+3\sqrt{3}\sqrt{-p^2q^2+4q^3+4p^3r-18pqr+27r^2}}}{3\sqrt[3]{2}}$$ and $$B=\frac{-p^2+3q}{9A}.$$  Then, $$x_1=-\frac{p}{3}+A-B,$$ $$x_2=-\frac{p}{3}+\frac{-1-i\sqrt{3}}{2}A-\frac{-1+i\sqrt{3}}{2}B,$$ and $$x_3=-\frac{p}{3}+\frac{-1+i\sqrt{3}}{2}A-\frac{-1-i\sqrt{3}}{2}B$$ are the solutions to $$x^3+px^2+qx+r=0.$$

