I don't see how Caesar prevented conquered Roman subjects from retaining their customs and traditions. Quite the opposite, actually. By and large Caesar's career had almost nothing to do with provincials, except for his conquest of Transalpine Gaul, which though horrifically bloody did very little to erase the customs of the Gauls. In Gaul Caesar continued the usual Roman practice of allowing local leaders to exercise power and except in cases where Gallic peoples revolted he really didn't bother too much with them, as long as they submitted to Roman rule. Unless we're supposed to understand that what is meant is that Caesar's provincial campaigns resulted in the mass slaughter of the conquered, whereas Augustus' did not. Which is pretty untrue--while the Gallic war was incredibly brutal, the Illyrian campaign under Augustus was equally nasty. I don't think there's really any comparison to be made here, Caesar had very little to do with provincial administration, it's not the way that the state worked until the provision of the imperium maius onto the person of the princeps

Did the school mean his conquered subjects (like the Gauls or some Germans) or the Roman people?

I have never personally heard it suggested that Augustus was better to the provincial people than Julius Caesar in that sense, anyway (he could be considered better in that he curbed the system of governors using provinces as their own personal piggy banks when he took ultimate control over them but that was never Caesar's goal or duty and would be an unfair comparison). Caesar was a brutal conqueror who killed on a fairly massive scale even for the time. But he also worked quite closely with Gauls and even some Germans who were already friendly to Rome. And he wanted some Gauls to be admitted to the Senate.

Allowing local people to keep their customs and traditions had long been a policy of Rome. Policy is probably too strong a word as Rome initially took a fairly laissez faire attitude toward empire building. But Rome had always adopted the local ruling class, local religions, and even local language when feasible (which is why Greek retained so much importance in the East) into the ruling structure. It is often considered one of the reasons Rome was so successful. Neither Caesar nor Augustus deviated from this particularly. Augustus made reforms to provincial administration and structure and took central control over. But that is really separate from whether they allowed locals to retain their customs (both did).

However, the comparison is often made when it comes to Rome, itself. You will often hear that Augustus was willing to allow the Roman institutions to stand and made his own personal rule subtle and less overtly like that of a "king." By contrast, Julius Caesar is often considered to have gone too far in overtly asserting his own right to sole rule.

That comparison is often misleading. Like Augustus, Caesar never actually did anything to dismantle or attack the dignity of the Senate. In fact, unlike Augustus, he went so far as having men who were against him and fought him in the Senate. And Caesar never took a title that didn't exist under the traditional Roman Constitution. He was a dictator. Which had always existed in the Republic. The six month time ban on dictators had been done away with before. There was the incident where Antony allegedly offered to crown him king three times. He refused it each time. Those who were uncharitable toward him took it to mean he would have accepted had the crowd been more receptive. But we don't know what he intended. And there is no reason to think he ever intended to be made king or anything completely against the Roman traditions.

Early on Augustus did have a title not in the Roman constitution. Under the Second Triumvirate, Triumvir became an official title of sole rule (shared of course). And Augustus, unlike Caesar, was initially brutal in executing and condemning anyone who opposed him. That he laid it behind him and kept up the ruse of the Republic doesn't negate that. He also later took the title of Augustus. And, like Caesar with dictator, the traditional titles he took were usually disproportionate to their old meanings. He became proconsul over most provinces for lengthy terms. He became tribune for life. Dictator does connote direct rule more obviously than the collection of titles that Augustus accrued. But I don't think anyone was genuinely fooled at the time.

Augustus benefited from that fact that the major faction of resistance had been brutally eliminated by himself and Antony through wars and purges. He had a much more pliant Senate before him than Caesar had. The Roman people had also had years of civil war by the time Augustus had defeated Antony and achieved solo rule. The benefits of continuing would have been minimal.

Augustus was more successful if the measurement is that he died of natural causes (unless you believe Robert Graves) as an old man. And there was a peaceful succession with his death. But, at the same time, Augustus was traveling down a path laid for him by Caesar. While it's true that he didn't have any high regard Caesar's name lent him more credentials than being Gaius Octavius ever would. And Augustus was never a great general the way Julius Caesar was. So, it is fairly unlikely he would have made a name for himself without the chaos surrounding the assassination of Caesar. I think it's more useful to see Augustus as building upon Caesar's legacy. Both bringing their unique talents to the problem of ruling the Roman world without claiming to be king.