The assumption that Constantinople consistently outpaced Western Europe in advancement is a historical oversimplification. While the Byzantine capital was undeniably a beacon of learning, culture, and commerce for centuries—preserving classical knowledge, pioneering architectural feats like the Hagia Sophia, and maintaining sophisticated administrative and legal systems—Western Europe was far from stagnant.
Different regions in the West experienced their own surges of innovation. The Carolingian Renaissance revived scholarship in the 8th and 9th centuries. The 12th-century Renaissance saw the rise of universities, the translation of Arabic and Greek texts, and advances in law and philosophy. Gothic architecture pushed engineering boundaries, and Italian city-states pioneered banking, trade networks, and early capitalism.
Constantinople's advantages—strategic location, centralized bureaucracy, and direct inheritance of Roman institutions—gave it a head start, especially in the early Middle Ages. But by the High Middle Ages, the gap had narrowed considerably. The Fourth Crusade's sack of Constantinople in 1204 dealt a blow from which the city never fully recovered, while Western Europe continued its trajectory toward the Renaissance.
In short: Constantinople was a singular hub of continuity and sophistication, but "more advanced" depends entirely on the century and the metric—military, theological, technological, or economic. The two worlds influenced each other far more than a simple hierarchy suggests.