So what’s the difference, other than one group believing that leadership of the religion after the Prophet’s death should go to his son-in-law and the other to one of his companions?

Depending on where you stand, not much because essentially they recite the same ‘Shahadah’ or affirmation of faith. But over the course of history, each group have evolved and follow their own practices. And over a thousand years, animosity between the two groups of Muslims have intensified and the battle of Karbala is repeated almost daily in strife torn countries like Iraq and now Syria.