The heros they bring forth will be one part Tiger Woods and one part Sammy Glick. The Fermat problem was solved only recently, almost a century after the announcement of Wolfskehl's prize. The competitive aspects of prizes can lead to publicly expressed acrimony. I know of three, perhaps four, serious instances of it during the years of my professional career. Recall that the Trojan war was begun because of the decision of a prize contest.
Large prizes can increase secrecy, bickering, and envy.
They could disrupt the sense of mathematical community. The heartbreak to the unsuccessful competitors is only equalled by the injury which their receipt can wreak on a weak or vain personality, or the irony of their reception by an aging scholar long after all good they can do is gone. I say, justly or unjustly administered, they are an abomination and should be abolished without exception. A million dollars is considerably greater than previously established mathematical prizes, and, according to a well-known dialectic that Marxists love to cite, quantity changes quality. To some, the Clay prizes spell a vulgarization of mathematics.