What do mean by "more advanced"? In terms of "technology", no, it was not. Indeed, there was very little change in military technology throughout Antiquity, and we must reject modern notions of "technological progress" when analyzing ancient armies. Remember also that the Roman army was a constantly evolving institution, changing as it encountered foreign peoples and different strategic environments. The army of Constantine was different than that of Hadrian, which was different than that of Augustus. So while I know what you mean, and will address it below, we cannot speak of a singular "old Roman army."

With that said, the army of Justinian was different than that of the West, though not fundamentally so. Over the course of the fifth century, the Western Empire, through territorial loss, became increasingly financially strapped. It could no longer afford to maintain large standing forces, and came increasingly to rely upon unincorporated foreign groups (a process further facilitated by civil war). The East, meanwhile, retained it's wealthiest provinces, and was more politically stable, suffering fewer civil wars. Constantinople was thus able to fund a large, professional army throughout the fifth and sixth centuries (though not without difficulty at times). As such, it is more apt to compare Justinian's army with that of the fourth century, when the military system functioned in both halves of the Empire.

Describing every institutional and tactical change over a two hundred year period would be a lengthy process, so I will focus on the most significant. The divisions between frontier-based limitanei and interior-based comitatenses became operationally and institutionally less meaningful. The basic tactual unit remained the regiment; infantry regiments being roughly 1,000 strong, cavalry, 300-500 strong. There continued to be differing internal rank structures, though it is possible to see increasing convergences. The basic panoply for the infantryman (and to a degree the cavalryman) remained unchanged, though the distribution of armor within a unit had, which I'll discuss in a moment. Most recruits continued to be native Romans (largely Greek speakers), though certain foreign groups, like the Huns and Heruls, continued to be targeted for recruitment, especially for their skills as mounted archers.

Tactically, there was a decisive shift towards cavalry. Cavalry, not infantry, was the arm of decision on the sixth-century battlefield. Although Procopius will laud the Roman cavalryman as being skilled with both bow and lance, it is clear that most cavalry units did not live up to this ideal. Although many units would have been armed with both, not every trooper would have been trained to use the bow. Most cavalry remained contact cavalry, able to both skirmish and engage at close quarters.

Infantry remained a vital component of the army, though it's tactical role had changed. Infantry became more stationary on the battlefield, acting as mobile refuges for cavalry, although they retained the ability to defeat enemy forces in close-order combat. Infantry also became more de-specialized, able to both skirmish and fight hand-to-hand. This tactical change is reflected in equipment. Armor became increasingly restricted to the front ranks, as only these under-officers would have found themselves in combat. Mail or scale corselets and greaves were worn only by the front ranks.

There are many other changes I didn't mention, especially in terms of dress, identity, and culture. I have focused on the institutional, as these are the most obvious. In 350 A.D., heavily armored infantrymen won battles through close-order combat, with cavalry playing a supporting role. By 550, it was the cavalry, well-equipped and highly trained, that was expected to be decisive. Roman infantry remained competent, but had assumed the supporting role. Both armies are equally effective in their own way, and illustrate how the army was always changing to new influences and circumstances.