So I can't answer your question perfectly, because my focus recently has been on the Roman army, before it technically becomes the Byzantine army (but then again that depends on when you believe eastern Rome became the Byzantines). I can give you a comparison between the early and late imperial armies though, and perhaps even give an indication about who was better.
The early Roman imperial army was comprised of two parts, which were roughly the same in size: the legions, and the auxilia. Both these forces were almost entirely stationed on the frontiers up until the reign of Septimius Severus (193-211), who was the first to concentrate a large force centrally, in Italy. This force was roughly 30,000 men strong (Durry 1938, 86). A legion had an approximate official strength of 5,120 men, and it was these men that you referred to in your question. They were heavy infantry, who fought in heavy armour with large, rectangular shields, and long spatha straight-edged swords, as well as the famous pilum (Southern and Dixon 1996, 91-116). They were commonly the 'core' of the army, fighting in the centre of the battle and seeing the heaviest fighting, and were also the primary siege assault force.
The auxilia were basically everything else. Their role was huuugely varied. They supplied almost all of the army's archers and cavalry, and even supplied some regular infantry units (although they were not normally as heavy as the legions Iguana_on_a_stick made it clear to me that this is not entirely true, and much of the auxilia were still heavy infantry, although different to the legions). They were also roughly numerically even to the legions.
Strategy also obviously plays a huge role in the effectiveness of an army. Like I said before, this army was almost entirely stationed on the frontiers for most of its existence. To form large campaigning armies, sometimes new legions would be raised, for example, for Trajan's Dacian Wars, although more commonly vexillationes would be drawn from legions on peaceful frontiers to form these large armies (Landelle 2015, 1066). This meant, at least for the time being, the entire Roman frontier could be guarded and large armies could be formed to either conduct campaigns or respond to threats without overly burdening the imperial treasury.
Yet the weakness of this system would be exposed during the Third Century Crisis. This involved massive destabilization, and almost constant civil wars between imperial claimants. This meant the armies were regularly being drawn away from the frontiers entirely to fight civil wars, exposing the frontiers to invasion would weakened the Empire further.
While a number of capable military emperors - from Gallienus to Carus - helped stabilize the Empire, it was not until Diocletian became emperor in 284 did the Crisis truly end. Diocletian made massive changes, dividing the Empire under the rule of two senior and two junior emperors. Each of these emperors was accompanied by their own armies - called the comitatus - to deal with threats. This was a significantly more stable situation - four emperors each with an army meant the threats could more rapidly and effectively be dealt with. It also greatly reduced the frequency of civil war.
Constantine I came along and in 324 reunited the Empire under his sole rule. HERE is where the most major change happens to the Roman army. This is the beginning of the late imperial army. In an effort to reward the troops that had been loyal to him and divide the troops which had opposed him, Constantine divided the army into three categories: the limitanei, comitatenses, and the palatini. The limitanei, about sixty per-cent of the army, were placed directly on the frontiers to protect against small-scale incursions (Janniard 2015b, 588). The comitatenses were mobile field armies stationed in the provinces. Not quite on the borders, but not in the centre of the Empire either (Janniard 2015a, 264). Finally, the palatini were placed directly with the emperor/s (Nicasie 2015, 720). In this new system the strategy for large campaigns was now for an emperor to move his palatini towards a threat, incorporating one or more comitatenses along the route, and then perhaps even utilising the limitanei in the campaign.
Gone were the distinctions of legions and auxilia. Now all of these forces were basically equipped like the infantry of the auxilia had been previously. They wore lighter armour than before, and used smaller, rounder shields. They were also shown to more commonly used spears rather than swords (Freshfield 1922, 87-104). The infantry were also expected to be able to use the plumbata, a kind of metal dart that was supposedly as effective as an arrow (Veg. Mil. 1.17; Arr. Tac., 44.1).
This was the basic nature of the Roman army up until the collapse of the western Roman Empire in 476. I have not mentioned the use of foederati and other barbarian units by the Roman army, because it is less relevant to the eastern Empire, and the actual extent to which it weakened the army is doubt-able.
So there are basically two ways to compare an army. Compare them fighting each other, and compare them in their time and place to see which was more effective.
If you sent the early Roman army in its entirety against the late Roman army in its entirety, I'd have to place my bets on the early army. The late army might have been slightly larger (Jones 1964, 347-80) yet the early army was absolutely designed for a pitched battle. Much like the modern military of the USA ten years ago was all tanks, planes, machine guns, and big ass bombs that could destroy any target anywhere, the early army was totally designed to meet an enemy in a pitched battle and end the day utterly victorious. While the late army was still excellent in a pitched battle, I feel that the heavy armour and weapons of the early armour would not be overcome.
In its context, however, this does not mean the late Roman army was any less effective. The late Roman army, and thus the early Byzantine army, was designed to be more fluid, more mobile, and more responsive. The weaknesses of the rigidity of the early army had been shown during the Third Century Crisis. If the army hadn't undergone the reforms it did, the Empire would have collapsed much sooner. Thus by comparison, both armies were effective in their circumstances.