I think it is a mistake to frame this as "republic with high moral standards" vs. "nasty backstabby empire." It's not helpful, it's not accurate.

The Romans certainly believed they were highly moral, disciplined, and virtuous, much more so than those "bloodthirsty barbarians." They believed this in the days of the republic, they still believed this in the days of the empire. (Though they sometimes instead believed that Romans USED to be moral a few generations ago, but that kids these days no longer respected their parents and were indulging in luxuries that made them weak. Tacitus, who wrote the Agricola, is one of the foremost and most sophisticated of these critics, so bear that in mind when you get to him.) But firstly this is the Romans being self-congratulatory as usual, and secondly their ideas of virtue were quite different from our own.

What your professor may be getting at is the changing demographics of the Roman army throughout their history, and the changing ideology surrounding it. This certainly was a thing.

In the days of the republic, the Roman army was a citizen levy recruited from a wide cross-section of the populace, but excluding the poorest, non-propertied class. Military service as considered a right and a privilege. Your position in the army depended on your wealth, as you had to own your own equipment: the richest would serve as cavalry, the well-off served as heavy infantry, and the poorer citizens would be lightly-armed light infantry skirmishing with javelins or slings.

The Romans at the time certainly espoused the belief that propertied men were the most reliable because they had a stake in the land they were defending, but it was mostly a practical matter: there were no state armouries, there was no standing army, so everybody had to bring their own weapons and the poorest of the poor couldn't afford any.

Soldiers would serve for a few campaigns at a time, usualy fought in the summers close to home, before returning to civilian life as farmers. (Though those who liked it, or younger sons who had no land of their own, might seek out more campaigns instead.) Warfare was a part-time, seasonal thing. Soldiers served for many reasons: pride in their social status, loyalty to their community, a desire for glory and adventure, and not in the least the hope of getting rich by looting and enslaving their enemies.

But make no mistake. The wars of the republic were extremely violent. When an enemy city was taken, the Romans would on many occasions engage in wholesale massacres or enslave entire peoples.

Then came new scenes of horror. As the fire spread and carried everything down, the soldiers did not wait to destroy the buildings little by little, but all in a heap. So the crashing grew louder, and many corpses fell with the stones into the midst. Others were seen still living, especially old men, women, and young children who had hidden in the inmost nooks of the houses, some of them wounded, some more or less burned, and uttering piteous cries. Still others, thrust out and falling from such a height with the stones, timbers, and fire, were torn asunder in all shapes of horror, crushed and mangled.

Nor was this the end of their miseries, for the street cleaners, who were removing the rubbish with axes, mattocks, and forks, and making the roads passable, tossed with these instruments the dead and the living together into holes in the ground, dragging them along like sticks and stones and turning them over with their iron tools. Trenches were filled with men. Some who were thrown in head foremost, with their legs sticking out of the ground, writhed a long time. Others fell with their feet downward and their heads above ground. Horses ran over them, crushing their faces and skulls, not purposely on the part of the riders, but in their headlong haste.

• Appian, the Punic Wars, describing the sack of Carthage

(Note: these dramatic descriptions of cities being captured and looted were quite common in ancient historiography, and indeed something of a cliché. But that only means they were common indeed.)

Note that the Romans did NOT see this kind of thing as unvirtuous. They would not even say the Romans were being particularly bloodthirsty here. Appian goes on to say

Nor did the street cleaners do these things on purpose; but the tug of war, the glory of approaching victory, the rush of the soldiery, the orders of the officers, the blast of the trumpets, tribunes and centurions marching their cohorts hither and thither - all together made everybody frantic and heedless of the spectacles under their eyes.

Later he describes Scipio weeping over the ruin of Carthage... but not because he pities his foes, rather because he fears such a fate befalling Rome.

So to the Romans, such violence was simly a fact of life, a matter of military strategy. The destruction of Carthage and Corinth in the middle republic were exceptional because of the unusual scale and thoroughness, and because the cities being destroyed had such a glorious and storied history, not because the destruction of a city was in itself terribly noteworthy. Caesar later boasts of killing a million Gauls and enslaving a million more in his Gallic wars. His fellow citizens take his figures with a grain of salt, but don't greatly fault him for it.

They DO fault him for breaking a truce and launching a surprise attack on the Usipetes and Tencteri. Massacring an entire people was not a stain on Roman honour. Breaking a truce was. (Cato the Younger argued that Caesar should be handed over to the the survivors so they could take revenge, but then again Cato was a long-standing enemy of Caesar anyway.)

So what change as the Republic transitioned into an empire?

As wars started to become lenghtier and occur farther from home, the seasonal part-time model of warfare became harder to sustain. It was more difficult to find enough volunteers for the legions, especially when campaigning in regions with less wealth to plunder, and conscription was unpopular. (Older historiography assumed that Rome suffered a manpower problem, with the propertied classes being eroded by the super-wealthy buying up their lands. Recent scholarship argues that the evidence does not support this, it's just a handful of elite writers who like to complain.)

The result is that the Roman army gradually transitions to a fully professional force, with soldiers serving for more than 20 years, and being equipped from state fabricae. (Though the cost for their equipment is still deducted from their wages.)

This professional Roman army becomes an entitity in its own right, with its own culture, particularly once it is stationed far out on the frontiers of the empire for generations at a time, far away from the urban centres of the Empire. (The western empire, that is. The in the east the army is mostly stationed in or near the great fortress cities of Syria and environs.)

But if anything, the professional imperial army gradually becomes less aggressive than the Republican army had been. Rome in later centuries fights fewer wars of conquest, and soldiers spend most of their lives in garrison duty, maybe fighting the occasional skirmish with raiders or going on a punitive expedition to avenge some real or imagined offense against Rome. In those punitive expeditions and in the wars of conquest the Romans did fight they could be as violent as ever, although there are no tales to compare to the destruction of Carthage and Corinth or the conquest of Gaul in imperial times.

For more information, an interesting recent book about at this topic is Spare No One: Mass Violence in Roman Warfare by Gabriel Baker, in which he argues that this kind of mass violence was an integral part of Roman strategy.