Why are gothic cathedrals considered the greatest accomplishment of the Middle Ages

Because it changed religion for centuries to come

And counties to do so today

Answer: as all great works of architecture cathedrals express spiritual or religious aspirations of people of that time. Cathedrals are specifically European phenomenon we cannot find elsewhere. Their elegant shape, narrow long windows and the play of light it produces inside is the best example of art of that time. The same can be said about the acoustic quality of their inner space. Cathedrals constitute a perfect combination of immaterial (spirit) and material (stone, durability).

Explanation: Cathedrals were usually built before the medieval world started suffering its crisis (14th-15th century) : especially crisis of papacy and purity of Christian faith. First marks of decadence start in philosophy (in fact few years after St. Thomas Aquinus's death) and with power conflict between kings (emperors) and papacy. Then there is a "Black death" (14th century) and schism in the Church. Cathedrals express human longing for the transcendent, for immortality a faith in almighty god ... something we lack today.