I guess that depends on a few factors. In regard to the answer below, Europe did have professional soldiers too. They were smaller in number and a very different system than what the Romans did, but there were men-at-arms and knights who were by definition, professional soldiers. That's not really the problem here at all.

The Romans could field massive armies. Truly massive, in the range of up to 150,000 men. The largest battles of pre-modern history were mostly fought in the classical Roman empire (despite Chinese claims of battles fought with 300,000 or 500,000 men, they did not have the logistics to even consider such a force, let alone their Xiongnu opponents. No society of the time could field 500,000 men.). Phillipi was fought with around 150,000 men on each side.

So your question is, why could the Romans do this when medieval Europe couldn't? Well first of all Medieval Europe could, the problem was entirely around maintaining a force that size in the field. Attila the Hun probably had an army of around 60,000 to 80,000 men at the height of the Hun empire, but also was taking advantage of short campaigns, high resource availability in his targeted areas of operations, and what he could get of Roman logistical knowledge and systems. The "Byzantine" Romans at Manitzikert probably fielded something like as many as 100,000 men, at least in what the sources state, but it wasn't impossible and they had the logistics to pull it off. Some estimate the first Crusade as approaching 100,000 soldiers. Some of the later medieval campaigns in the 14th century start approaching larger sizes like this as well.

The problem was that fielding an army over about 25,000 men was nearly impossible. And this doesn't just apply to medieval Europe, but to the Romans as well. You're talking about marching columns that even on the widest Roman roads, would be arriving at their destination at the same time the rear of the formation was leaving where they had camped the night before. So these armies had to be split up, or extremely mobile, in order to be fielded in those sizes of 50,000 or 100,000 men. Not to mention equipping and feeding those armies. The Romans had a highly advanced logistical network of granary depots, arms resupply stations, and horse/mount resupply stations along their road network. A bureaucrat using the cursus publicus, which was the system set up for officials (not the military system), could cover 80 miles a day thanks to it, rather than 20 miles a day (overland, of course). They had an advanced system of of administration that was able to manage the flow of money and resources into the government, and redistribute that back out to the depots and directly to the soldiery. They also had laws on top of that which taxed local citizens to supply their local forces namely with items in kind (although sometimes in cash to purchase items), namely clothing. And even then, it took months or even years to prepare for a campaign. Constantius II had to send supplies to the Pannonian frontier for nearly 6 months to prepare for his campaign against the Sarmatians. Tonantius Ferreolus spent 2 years preparing for Attila's invasion of Gaul.

Medieval Europe simply did not have this. In most situations soldiers were responsible for their own supplies. They didn't even have senators paying them out of their own pockets like the Marian army or the Polybian army of the Roman Republic, which covered the cost of much of the logistics and allowed the Romans to field massive forces without a professional system. Medieval Europe did not have the upper-level administrative capability to manage armies of that size, nor the physical infrastructure to transport those supplies to them. The exception to this is by and large the Islamic world, China, and the Crusades, the last of which was heavily supported by the Romans, who used their logistical system and navy to supply and support the army as it passed through the Balkans and Anatolia, and even then it wreaked havoc on the landscape. And this is because these civilizations had the upper-level organization and were able to employ systems of manufactuories and depots to prepare for and manage these operations.

If you really want to get into the nitty gritty of how the Romans organized and supplied their armies, there's an excellent book on it by Jonathan P. Roth called The Logistics of the Roman Army at War. It's a free PDF online somewhere if you can find it.

I think the comparison is a little flawed. Rome and Carthage had far greater populations and resources than medieval Britain. Bosworth was relatively small scale even by the standards of medieval Britain, as Richard III had decided to strike before either side could gather reinforcements. Two decades before Bosworth, the battle of Towton had 30,000 men on each side. Julius' Caesar's army was around thirty thousand men strong for most of his Gallic campaigns.

As u/FlavivsAetivs says, it was largely an issue of logistics. Medieval Europe could gather large armies on occasion, but it didn't have the infrastructure to keep them supplied, and large armies were unwieldy. It was far more practical to organise your society to produce a small number of highly trained fighters, who you could then keep supplied for a few months campaigning, than conscript every peasant who could hold a spear.

There's also a very obvious point that no one's mentioned: medieval states were far smaller than the Roman Empire. They simply had less manpower to draw on in the first place.