The basic difference between the world (in Europe, at least) of those two years is that the world of 1250s was a medieval world while the world of 1450 had already entered the Renaissance.
There are many other differences, some of which contributed to the changed outlook. For example, the world of 1450 had experienced the Black Death. It had also been through the era in which there were two or more men at a time claiming to be pope. These sorts of things had changed the mindset of some Europeans, moving them towards humanism.
Humanism had become a major idea by 1450. Petrarch and Dante had written their major works. Early Renaissance artists like Giotto and Donatello had changed the face of art. Because of this sort of change in attitudes, the world of 1450 was very different from that of 1250.
The world of 1250 was quite different from the world of 1450. Europe's population in 1450 had yet to rebound from the Black Death of the fourteenth century, whereas Europe in 1250 was growing due to a climate warming trend which allowed for a longer growing season. In 1250, Europeans still referred to themselves as "Christians" first and then members of a nationality second. By 1450, the idea of the nation-state was starting to take hold. Christianity was still quite strong with Catholicism as the exclusive doctrine, but an idea of secular power was starting to take hold by 1450. Archers and siege engines (catapults and battering rams) were the keys to a successful war campaign in 1250, while by 1450 gunpowder technology was starting to change the face of war in Europe, much to the chagrin of Constantinople in 1453 when it fell to the Turks. Europe in 1450 was more interconnected by roads and trade routes, and there was no fear of Viking or Mongol invasion, unlike the Christendom of 1250 that was always under attack by non-Christians such as the Vikings, Saracens, or Mongols. Also, by 1450, the Crusades were over, and Europeans started to look for more ways to reach Asia for spices, not to convert Asians to Christianity.