The history of Western Europe would undeniably be very different had the Frankish army been defeated in 732. Like the victory of Constantinople some fifteen years earlier, Muslim advances had finally been thwarted, and the steady Reconquista of Spain and parts of Byzantine Empire began apace.
Though the Byzantines finally succumbed by 1453, the Turks were eventually defeated in the great naval battle at Lepanto in 1571, and turned away from Vienna in 1683.
The threat of the Caliphate encroaching on Western Europe never penetrated as far west again--until today, as unassimilated Muslim immigrants threaten to "behead those who insult Islam" and foment rage at every perceived slight.
A most excellent analysis! MAJOR KUDOS FOR DUE DILIGENCE!