Yes, for a number of reasons.

From the early to mid Republic, armies consisted of the landed classes. Property was a requirement for service, and your rank and troop type (hastati, short-ranged swordsmen similar to legionaries, velites, javelin armed skirmishers, and triarii, spear-armed veterans usually of more advanced age, were the primary variants) depended on how much wealth you could afford to spend on arms and armor, or how much prior experience you had as a soldier. Gaius Marius reformed the military and disbanded the property requirement, in effect turning the Roman army into a full-time, professional standing army. Citizens from all walks of life could join up and serve, and when legions were not on campaign, soldiers were subjected to regimens of physical training, arms training, and disciplinary drills.

With the adoption of the cohort, the military itself transformed into a highly organized machine. A legion normally consisted of around 5,000 men broken down into ten cohorts, each consisting of about 480 men. Cohorts in turn were broken down into six 80-man centuriae (centuries), each commanded by a centurion. The first cohort of the legion, usually the most important as it possessed the eagle standard of the legion, had closer to 800 men broken into five double centuriae (160 men instead of the usual 80). The military hierarchy consisted of a legate (commander, usually of senatorial rank) at the top, military tribunes (of both equestrian and senatorial rank) below, and centurions at the bottom. Centurions were responsible for directing their centuries in battle and disseminating orders to their troops. Because of them, Roman armies moved and operated as a coherent whole. Evidence exists that suggests the possibility that centurions could often direct the line of battle and maintain order even in the absence of quality generalship.

The armies or military forces of other peoples, such as the Parthians, the Germanic tribes, the Illyrians, etc. often lacked the organization, professionalism and coherence of the Roman army, let alone the discipline that enabled the legions to have as much success as they did.

Tl;dr: Yes. Roman discipline and professionalism were hallmarks of their military organization and far eclipsed that of their enemies.

In terms of sources, I recommend:

The Roman Army at War: 100 B.C.-A.D. 400 by Adrian Goldsworthy

Agricola, Germania and The Annals of Imperial Rome by Tacitus

Conquest of Gaul by Julius Caesar All offer good insights into the character and responsibilities of the Roman army.

I can't speak for other contemporary ancient armies. However, the Roman army had a rather brutal practice indeed: that of decimation.

When Roman soldiers disobeyed orders or retreated, they were subject to decimation as punishment. If you consider the word itself, you'll find out what decimation is about. 'Deci-' as a root often means ten; the punishment begins each member of the Roman legion in question getting together with his group of tent-mates (contubernales) of 10.

The ten men then draw lots. Whoever draws the proverbial short straw then is beaten to death by the other nine men.

This punishment was used famously by Crassus, of the famous first triumvirate, when his subordinate Mummius disobeyed him and lost nearly two whole legions. This is detailed in the Life of Crassus by Plutarch.

The Complete Roman Army by Adrian Goldsworthy has a bit more to tell us on decimation:

The remaining 90% of the unit...were ordered to set up their tents and sleep outside the rampart of the camp, and were issued barley instead of wheat. (p.101)

A bit more from Goldsworthy on general punishment:

The vitis, the centurion's vine cane, was frequently used to inflict beatings, leaving scars on many a soldier's back. This corporal punishment appears to have been inflicted entirely at the whim of these officers. (ibid)

Sources:

Polybius, Histories, Book VI.38

Plutarch, Life of Crassus

Adrian Goldsworthy, The Complete Roman Army

edit: added a few more sources & info, cleaned things up a bit

A new Roman recruit, after passing the medical examination (probatio), would go through basic training that consisted of physical fitness, marching in step, and keeping formation. The recruits would go on regular marches to improve their stamina, 30 km in 5 hours at normal pace, or 36 km at quick pace. Weapons training was the same system that was used in gladiator schools, first using a wooden sword and a wicker shield that were heavier than normal equipment, so that while practicing cuts and parries they would also build up their strength.

After some time the level of training would increase and they would fight mock battles with either practice weapons or real weapons covered in leather discs to prevent injuries. It would start with soldiers fighting in pairs, and then groups, and then it would move up to entire units.

Training would continue throughout the soldiers' time of service, and Josephus noted that 'it would not be wrong to call their drills bloodless battle and their battles bloody drills' (Bellum Iudaicum), or at least that was the ideal.

Like many other armies in history, the Romans tried to keep the army busy, as there was nothing more dangerous than a group of soldiers with nothing to do. A small amount of records of the massive bureaucracy survives, and from it we know of typical tasks that might be assigned to a soldier. Some would be assigned guard duty at the walls or gates, patrols around the fort, then there was repair and care of footwear, artillery, running and maintenance of the bath house, wagon repairs, arms maintenance, street cleaning, a range of administrative duties, and others.

The system of punishment has always been severe, from the days of landed citizen army of the Republic to the fully professional army of the Principate. For lesser offenses, the punishment would be flogging or a fine. A soldier who committed a serious offense would be subject to court-martial by the military tribunes. Fustuarium (being clubbed to death by your comrades) was administered for dereliction of duty (like falling asleep on guard duty), thievery, bearing false witness, for 'one who in full manhood is detected in shameful immorality', and being caught three times in committing a same offense (Polybius, Histories, 6.37).

If a larger number of soldiers committed the same offense (like cowardice), the punishment would be decimation. This is what Polybius says about it: 'The Tribune assembles the legion, calls the defaulters to the front, and, after administering a sharp rebuke, selects five or eight or twenty out of them by lot, so that those selected should be about a tenth of those who have been guilty of the act of cowardice. These selected are punished with the fustuarium without mercy; the rest are put on rations of barley instead of wheat, and are ordered to take up their quarters outside the vallum and the protection of the camp.' (Polybius, Histories, 6.38) The system of punishment was a cause of both mutinies and desertions throughout the centuries.

On the other hand, if a soldier performed acts of exceptional bravery, there was a system of military decorations and public acclaim to reward them. First a consul would summon an assembly of the legion, and call forward any that distinguished themselves. Then he would individually compliment them on their actions, and list all other distinctions that they earned in their career, and then present them with a reward. These acclaims were a source of great pride for the soldiers and as Polybius says 'for the recipients of such rewards not only enjoy great glory among their comrades in the army, and an immediate reputation at home, but after their return they are marked men in all solemn festivals; for they alone, who have been thus distinguished by the Consuls for bravery, are allowed to wear robes of honour on those occasions' (Polybius, Histories, 6.39).

Listing and explaining all the military decorations would be difficult , but here are some of the most well-known ones:

corona obsidionalis - Awarded to the man responsible for saving a whole army or city. It was made out of grass, for symbolic reasons, and which is why it is also known as corona graminea. It was the most difficult one to earn. Because of the circumstances, it was mostly awarded to officers and commanders, and the lowest ranked soldier that it was awarded to (that we know of) was a primus pilus centurion Cn. Petreius, in the Cimbri war. It was famously given to Fabius Maximus by the Senate for saving the city and state during the Second Punic War. It is uncertain when this award exactly fell out of use, though there is reason to believe it fell out of use before the end of the Republic.

corona civica - second only to the grass crown. It was awarded to a man who, in battle, saved a life of a Roman citizen, and held the place where this has occurred. Furthermore, the citizen that was saved had to attest to this and witness testimonies would not be taken into account if the saved person did not acknowledge this. The man who won this award would be entitled to wear it his entire life. He had the right to sit next to senators at games, and they would be required to stand up at his entrance. For the rest of their life, there would be a father-son sort of a relationship between the rescuer and the rescued, as mentioned by both Polybius and Cicero.

corona navalis - awarded for victories at sea.

corona muralis - given to a soldier that is first to scale the wall or enter by force into an enemy town. It likely wasn't awarded very often, as being both the first to scale the wall and surviving would be difficult.

corona vallaris - counterpart of the corona muralis, but awarded for being the first in enemy camp rather than city.

corona aurea - golden crown, for rewarding gallantry of type not specifically covered by other ones.

vexillium - Flag. Unclear when it was first given out. It is included in the honors Marius has won in 107 bc, and it was given to Agrippa in 31 bc for a naval victory over Mark Antony at Actium. Became a standard award for officers in the Principate.

armilla - Although bracelets were considered to be an article of women's clothing in Roman society, armillae or military bracelets were exception to the general rule. Like torques, it is likely that they adopted the custom of wearing them from Celts. During the Principate, they were never awarded to an officer of a higher rank than centurion.

This is a highly complex system of public punishment and shame, and public awards and acclaim. The whole concept is hardly unique to the Romans, but they were very systematic about it.

Sources:

Polybius - Histories

The Complete Roman Army - Adrian Goldsworthy

The Military Decorations of the Roman Army - Valerie A. Maxfield