In short, because only positive numbers could be used as coefficients at the time there were several cases of the cubic that had to be treated separately. Before Cardano only depressed cubics (with one of the powers missing) were solved. He introduced the substitution that reduces general cubic to a depressed one, and made formulas for depressed ones public for the first time, with his own derivations. Ironically, the formula under his name is not due to him, but he was the first one to "publish" it, while acknowledging del Ferro's and Tartaglia's contributions. Since the use of complex numbers was rudimentary at the time even Cardano's generalization did not resolve the case of a cubic with three real roots, the irreducible case, as Bombelli pointed out. It was later resolved by Viète, who related it to the angle trisection problem.
Cardano's masterpiece, Ars Magna (1545), was the most comprehensive treatise of algebra until Viète's Isagoge (1591). Unlike del Ferro and Tartaglia who dealt with special cases only, Cardano provided systematic treatment based on substitutions of all thirteen cases of depressed cubics, and of the general case. It is no accident that it was his student, Ferrari, who solved the quartic following the same methods (his solution was also included into Ars Magna). Moreover, Cardano acknowledged the possibility of negative ("fictitious") roots, and in one example considered complex ("sophistic") roots for the first time. Ars Magna's influence is often compared to that of Copernicus' De revolutionibus, and it did a lot to promote new methods in algebra at the time. Most mathematicians learned the cubic solution formula from it, which explains the naming. This is similar to "Pascal's" triangle. It was known even in Europe long before Pascal (curiously, Tartaglia claimed it as his), but Pascal was the first to systematically study its properties.
The discovery itself is a fascinating story. It was Scipione del Ferro who solved the first depressed cubic. At the time it was common for mathematicians to call each other to "duels", where having a secret trick for solving equations no one else could solve was a big advantage. These duels often carried monetary prizes, and affected recognition and appointments in ways publications do today. So del Ferro kept it a secret, which he entrusted to his student Fiore. Tartaglia challenged Fiore and shortly before the contest solved del Ferro's case and one other (he admitted though that what helped him solve it was knowing that solution existed). Needless to say he won the duel.
After futile attempts to solve cubics by himself Cardano convinced Tartaglia to give him the formula with a promise that he would keep it a secret until Tartaglia publishes it in a book he was writing at the time. However, Tartaglia took his time and eventually Cardano found out that del Ferro discovered the formula before Tartaglia. He no longer felt bound by the promise, and published the formula in Ars Magna. Tartaglia was understandably livid. The coda was Ferrari challenging Tartaglia to a duel after discovering a solution to quartics. That one Tartaglia lost.