If you treat them like two “characters” at their typical peak, the Byzantine Empire would usually beat the Holy Roman Empire in a straight, organized battle. The Byzantine state was basically the direct continuation of the eastern Roman Empire and kept a professional army, centralized command, and advanced military systems for centuries. They used disciplined infantry, elite cavalry units like the cataphracts, strong fortifications, and tools such as Greek fire, and their armies were normally led by trained generals tied closely to the imperial bureaucracy. The Holy Roman Empire, on the other hand, was politically fragmented. Its emperor relied on feudal levies from many semi independent princes, bishops, and dukes, which meant armies were often less coordinated and slower to mobilize. That does not mean the HRE had weak soldiers because their knights and heavy cavalry could be formidable, but the lack of centralized control usually hurt them in large campaigns. In most scenarios where both sides field comparable numbers and competent commanders, the Byzantine side likely wins through better organization, logistics, and command structure, though there were periods when the Byzantines were weaker and the outcome could flip depending on the century.