Contrary to popular belief . . .
I learned of this a few years ago upon reading Antonia Fraser's captivating biography, Marie Antoinette: The Journey. In it, she briefly espouses on the popular myth, explaining that the phrase was actually uttered 100 years before Marie had even been born by Marie-Thérèse, the wife of Louis XIV. Fraser continues, "It was a callous and ignorant statement and she, Marie Antoinette, was neither.
Another important note: The phrase was printed in Jean-Jacques Rousseau's Confessions, the final volume of which was completed by 1769 when Marie Antoinette was just 14; it would be another year before she even arrived at Versailles to marry the Dauphin.
Finally, as for Let them eat brioche versus Let them eat cake? My best guess is that it's just how the English language has adopted the phrase. I'd be curious to know what they say in France; is it still Qu'ils mangent de la brioche, or has it altered to Qu'ils mangent de la gateaux? Anyone know?