I can not speak for medieval times, but I can speak somewhat for antiquity in the Mediterranean generally.

When considering how armies are provisioned with equipment, there are a few things that one needs to take into account; the social background which determines who supplies what to a soldier, the fighting tactics used by the army, the size of the army, and the cost. For say, an Athenian hoplite army, generally the tactic was fighting in groups, so a large shield was desired that could cover yourself and the man next to you; the soldiers provided most of their own kit, armies were not excessively large, and most people were heavily armored. The army was generally used in “single battle wars,” where you’d go out in the summer, fight a battle to decide something, and call it quits to get back home and farm again. This is very different from late Roman armies, which are a professional force, provisioned by the state, which need to be highly mobile to deal with hit and run tactics utilized by their enemies, to police the borders, and to have a diversified unit structure with many different roles for different uses.

As far as I can surmise, armament changed along the needs and uses of the social context in which the army was raised and maintained. When Athens needed more troops in the Peloponnesian War, men were more lightly armored in order to allow people of poorer status to serve in the military, and likewise many were put into naval service which required no armor. When the roman state faced a fiscal crisis in the 3rd century, it was desired to make military equipment cheaper, doing away with much of the heavier basic equipment as it was expensive to produce such as Lorica Hamata, or chain mail. In a similar vein, the chain mail of the Romans was much more time intensive to make than later, say Norman chain mail; the rings of Roman Lorica Hamata were made of generally smaller riveted rings, and thus more had to be used in the making of the lorica than the larger rings used to make hauberks in the medieval period. Some reenactors argue that the smaller rings used in Roman armor were superior, and that this might be reflective of the Roman Empire’s wealth compared to medieval states; the people using medieval chainmail with larger rings, however , seem to have had little desire to go back, and their chainmail seems to have been effective enough for them.

I would argue, generally, that to say a certain type of equipment was “improved” from what period to another is completely relative; we might argue that one style is factually more effective today, but the fact remains that of the materials and artifacts we have remaining to examine today, many were used in battle, and thus people felt comfortable using them in that context to some degree. It didn’t matter to a medieval knight that the Romans used chainmail, and that it was better (though he likely never would have heard such a thing voiced); what matters to him is that his mail is good, clean, and will keep him safe, and is the best equipment he can afford to use given his situation. The same could be said for a Roman centurion, or an Athenian hoplite. Some method existed for them to get their hands on armor and weapons, and they used those to arm themselves as best they could.