The disparity between Medieval army size and Classical/Antique army size really is down to one major thing, state capacity, and I'm sorry if that's not an answer that you find compelling or sufficient, but it really does boil down to the nuts and bolts of military planning and supply, or logistics.

The medieval world was filled with states that were just simply not able to muster the same amount of men under arms because of the constraints that medieval armies and societies faced compared to Roman, Hellenistic, and other Classical world states because they lacked the administrative capacity, the levels of urbanization, and coercive political systems that the states of Antiquity enjoyed. However this level of uneven army size is mostly applicable to western Medieval Europe, nor was it constant.

Let's start with the basics, as the Western Roman Empire disintegrated over time its institutions and bureaucracy likewise collapsed. The ability of the Post-Roman "Barbarians" to maintain Roman systems such as the large standing army (which was already much smaller than its heyday some centuries before), large urban centers, and efficient (relatively) tax collection was extremely limited. The collapse of long distance trade, the Roman bureaucracy, and the diminished ability of post-Roman, and even post-Carolingian states to effectively administer their territories, meant that actually getting tens of thousands of individuals into a single army was a logistical nightmare. Likewise, the same reduction of capacity in logistics and administration meant that the ability of Medieval governments to pay and feed such large hosts would have been doubtful at best. The Romans did not have these same limitations, by being able to call upon the vast tax revenue of the empire, and requisition supplies from all around the Mediterranean, the Roman army could put together, and sustain (and this is the really important part) large armies on a permanent basis.

I'll just link /u/Celebreth's excellent answer on Roman logistics so that you can get a sense of just how large and involved the Roman logistical machine was.

Check it out here

Now why did this decline in state capacity occur in Late Antiquity and the Middle Ages? After all as you note, other polities such as the Britons and Gauls were able to raise very large hosts as well, and they too lacked the Roman penchant for logistical organization. Now we can chalk some of the huge number of men under arms in Germanic, Gallic, and so on army sizes to the hyperbole of ancient authors, but even taking this into account it does appear to be the case that even less urbanized and bureaucratic states were able to leverage a larger amount of their population into soldiers. The major difference here is going to be that Medieval states had additional layers of what I will call fractured authority.

The king of France in the year 1000 was theoretically in charge of all of France as one would assume, however, his practical power extended a few tens of kilometers around the city of Paris and that was about it. The lords of Normandy, Southern France, Aquitaine, and more functioned essentially as independent figures, and the idea that the king could exert effective coercive power over their economies, military, logistical networks, and the like was laughable. This was the case in much of Western Europe. As Roman social organization, urbanization, and the like broke down, local powers, what we might place as the "nobles" of the "Feudal system" stepped in, however these lords came with their own interests, quarrels, and power bases which precluded most post-Roman Western European states from exerting effective control fully over their own nominal territory. Throw into that mix the utter lack of maintained infrastructure like roads, supply depots, and all the other things that make huge armies possible, and you've got a series of issues that were plaguing any would be large Medieval army from the outset.

This was also heightened by other factors as well. The Medieval World's armies were not like those of the Roman or Hellenistic worlds. Most Medieval armies were not made up of, contrary to popular depictions, of rabbles of peasants given a little training and sent off to fight. By the time of the High Middle Ages, around the year 1000 or so, the Medieval armies of groups like the Normans, Anglo-Saxons, Byzantines, Italian States, and so on were rather professionalized. By the time of the 100 Years War in France, the armies of France and England were thoroughly professionalized that drew on large pools of mercenaries, wealthy urban dwellers, and other manpower pools. So why weren't these armies the same size as armies of Antiquity? The answer is rather simple really, cost. The armies of the Middle Ages were relatively more expensive to the states that fielded them than the Roman Empire's armies. Given the vast logistical network of the Roman army and the incredible financial resources the Roman state could rely upon the small size of Medieval armies makes a lot more sense. Now throw into that other elements, the much more expensive demands on individuals for Medieval soldiers for their equipment (Medieval Europe lacked the capacity to provide uniform equipment, including weapons and armor to their soldiers), the reduced ability of the state to adequately fund large armies over a long period of time, and again the answer becomes rather clear, it is all down to state capacity and logistics.

Despite these limitations, there were still states that were able to assemble large armies from time to time. The English under the Anglo-Saxons maintained a system called the fyrd that could be called up as essentially a mass levy to fight off attackers, but the state still lacked the capacity to maintain them as a standing force, and even when called up they could only be fielded for a certain amount of time, they would eat themselves to death otherwise. Now in some cases, such as in the Byzantine Empire, the various Islamic states, or during the Crusades much larger hosts were assembled, Thomas Asbridge for example estimates the strength of the First Crusade at over 100,000 people (many of whome were not combatants, knights, soldiers, or fighters really) when all is said and done, but they were not one cohesive army, they couldn't be really. The logistical nightmares facing Medieval armies really come home in looking at the First Crusade. It was to be frank an embarrassing display of Western Military competence at the time.

Because if its disparate and disorganized nature the crusaders were barely able to keep their army together, and indeed at times utterly failed to do so! The followers of Peter the Hermit for example were nearly all killed or enslaved when they arrived in the Holy Land too early, much of the Eastern European countryside was subjected to looting by the crusading forces, they consistently ran out of food, and were such a shambles at time that it is honestly a miracle they managed to seize Jerusalem at all. Their armies often fought divided, bickered over how to conduct sieges, arrived at different staging points at different times, and all of this was probably still the best that they could actually manage to do.

Keeping large armies fed and cohesive has been a challenge in all states that field armies, and there are always draw backs to different approaches. In the Middle Ages, the collapse of state authority compounded these problems, and it was only later on in the Middle Ages as state capacity rebuilt itself through renewed urbanization, the creation of a new bureaucracy, and the gradual consolidation of power in royal figures that army sizes started to consistently approach the sizes of Roman armies at the empire's height.

There are a few reasons. u/Steelcan has already discussed the big one - logistics. I'd like to go into some of the other issues with medieval vs. classical armies, including the premise of the question.

All estimates are dodgy.

But I do have to start with a caveat: all numbers are ballpark estimates. Few classical sources can be trusted with their numbers for armies, and it's the same with medieval sources. No historian from these time periods regularly sat around counting every man or ship or horse, they're guessing and so are we. There are a couple of assumptions that get made though, particularly outside of academia. Firstly, there is sometimes an assumption - especially with Roman armies - that they fought at full strength; that legions were fully manned when they went into battle. That is disputed. Secondly, there is sometimes an assumption that medieval armies were smaller on the basis of logistics that isn't necessarily true. For example, the lowest contemporary account of William the Conqueror's invasion force puts it at 14,000 but some historians think it was as low as 7 or 8 thousand. I'm not sure there's really anything wrong with William the Conqueror having an army of 14,000 if we can also believe that Robert Guiscard had an army of some 15 or 16 thousand for his invasion of the Balkans in 1081. It's all estimates and they can be out by thousands per side, but I do think there might be a tendency to overestimate the size of classical armies while underestimating the size of medieval ones and quite frankly there is no solidly reliable methodology behind estimating medieval army sizes. A big issue with both classical and medieval accounts is that it's usually very unclear who is actually being counted in the army. Just legionnairs or also auxiliaries and support staff? Just knights and their households or also mercenaries? It's usually unclear and accounts can easily leave out thousands of people who fought, or include thousands of people who didn't actually fight at all.

In theory, medieval battles could feature over 100,000 men with great difficulty.

The largest battles of the Middle Ages featured as many men as major classical battles. The Battle of Dorylaeum between the First Crusade and the Sultanate of Rum in 1097 likely had over 100,000 people on the field by its end when the bulk of the crusading army arrived (though the crusading army was itself a coalition of smaller armies). The Battle of Manzikert in 1071 between the Seljuk Empire and Byzantine Empire also may have had over 100,000 depending on whether we count the half of the Byzantine army that vanished in a puff of smoke earlier in the day as being part of the battle. The mobilisation of armies of 50,000-100,000 men was certainly possible, but only in massive military campaigns on special occasions involving major empires or coalitions of smaller armies, such as the crusades. And the strain that it put on those armies was biblical as u/Steelcan has explained. It was a mammoth undertaking to put together the scale of armies once mustered by Xerxes or the Roman emperors, but it could be done.

Medieval battles were smaller because the belligerents were smaller. To scale, medieval battles weren't much smaller.

When this question is asked, there is usually - and this one is no exception - a comparison between a massive classical war (in this case the Second Punic War) and a regional medieval conflict (in this case the Wars of the Roses). Why wasn't the war between two noble houses as large as the battle between two mighty empires? Well, because the former was fought by a few noble houses while the latter was fought between two mighty empires!

Most medieval wars were fought on the scale of individual counties. A duke might fight a count over a territorial dispute. A viscount might fight another viscount over the inheritance of a castle. Even the most powerful noblemen - men like the Duke of Aquitaine who ruled much of southern France or the Duke of Apulia, Calabria, and Sicily who ruled Italy south of Rome - had limited resources more akin to a single Roman province or a couple of Greek city-states than an empire like Carthage or Rome or Persia.

As it's a bit unfair to compare England to Carthage, let's scale things appropriately. Let's look at the classical and medieval versions of a France (or Gaul) based warlord invading Britain. In 55BC, Julius Caesar invaded Britain with a force of two legions and auxiliaries, transported by over 100 ships. Those forces are very unlikely to have been at full capacity after years of campaigning in Gaul (Caesar's first invasion was at the end of the campaigning season when his manpower would probably have been at its lowest in the year) so it's probably around 8000-10,000 men. The second invasion a year later after this first one went awry involved five legions with auxiliary cavalry, so probably around 25,000. The transport fleet was 800 ships. Our sources for William the Conqueror's invasion of Britain suck, but they suggest a size of invasion force somewhere between Caesar's first and second attempts, albeit on the lower end. William's fleet was apparently 776 ships, but there is no way of knowing how accurate that is and I seriously doubt anyone bothered to actually count them all. We do not have reliable figures for the number of men and historians vary wildly in their estimates. On the lower end are estimates of 7-8000 men, though I personally don't find that persuasive given this army could overcome the numbers generated by the English fyrd system. But even that lower estimate puts it at the same level as Caesar's first invasion. Higher estimates go up to about 12,000-14,000 men, which would land it nearer the middle of Caesar's two attempts. So when we compare like for like, the medieval armies are not necessarily much smaller.

Ditto invading Sicily. In 415BC (and there are people like u/Ikphrates on this subreddit who know orders of magnitude more than me about this so please correct me if I'm wrong) the fateful invasion of Sicily by Athens only involved a few thousand hoplites and light infantry, not more than 10,000 in total to start with. When the Romans invaded Sicily and conducted the famous Siege of Syracuse in the Second Punic War, their army was seemingly just under 20,000. When Robert Guiscard invaded Sicily in the 11th century, his army was drip-fed into the region so it's hard to gauge, but when he invaded the Balkan side of the Adriatic Sea in 1081 he apparently had 16,000 men and that was after many years of war in Sicily. If we stop comparing army sizes between vastly different entities and instead compare campaigns of similar scope, the differences in army size often melt away. It seems that both the medieval and classical world were capable of marshalling similar army sizes for limited campaigns. It's only when the biggest of wars happened that the medieval world falls behind, which is to be expected considering that the Emperor of Rome and the King of Leon are working with very different levels of resources.

Medieval military doctrine strongly disliked risk.

The all or nothing battle where one's entire forces were piled into the campaign was insanity to medieval commanders. Keep in mind what I said about the limited resources of medieval nobles. If the Roman Empire lost an army of 20,000 they could replace that loss by transferring legions from elsewhere. If Robert Guiscard lost an army of 20,000 he was completely fucked. The major battles of the Middle Ages were almost all disasters for one side or another. For example, the Battle of Bouvines in 1214 (France vs. England, Flanders, and the Holy Roman Empire) was unknown in size - estimates vary from less than 20,000 on the field to about 100,000 - but a battle where both sides committed to a classical style big battle on the open plains. It was a perfect demonstration of why that was a terrible idea in medieval warfare as France won against all-comers, capturing or killing over 300 knights and thousands of infantry that (again keeping in mind limited resources) were difficult to replace and impossible to replace within the same campaign season. It had major repercussion in England as the royal army was so weak that rebel barons who wanted a charter of reform began mobilising for the war that resulted in Magna Carta. John learned the hard way that committing one's forces to a major battle was a gamble not a strategy.

The military authority bar none in the Middle Ages was the late Roman author Vegetius. His advice to his readers was really quite simple: do not fight big battles on open fields, use Fabian tactics or small cavalry units until you win or go home. Most medieval warfare had nothing to do with big battles. Warfare was fought by raiding the territory of the enemy or by trapping them in a castle for a siege. As much as medieval literature glamourised the clash of lances in a mighty cavalry charge, most knights only fought two or three battles across their entire lives. It's hard to overstate how risk averse most medieval leaders were, and many medieval treatises on warfare stressed that once the lines clashed the result was as much up to chance and God as it was to strategy and leadership. Most were careful to only commit forces when they had gauged the terrain and had a plan - Richard the Lionheart had a bespoke plan for almost every battle, and only at the Battle of Arsuf on the Third Crusade did he really make the gamble of committing his forces in full (and refused to march on Jerusalem afterward because Arsuf kinda ruined his plan).

This question pops up every few weeks. This post by /u/FlavivsAetivs is frequently cited, though of course more can be said.