It has been argued (and quite convincingly, to my mind) that the main reason the imperial system was able to continue was that Augustus lived for so long. By the time he died, he had been sole ruler of Rome for 41 years. The vast majority of the population in AD 14, when he died, had never known anything different. The days of Cicero's Republic were long gone. Even the oldest members of the senatorial order, those who grew up alongside Augustus himself, had no experience of the political world before Caesar's rise to power.
I suppose though that that only answers part of your question. The reason Augustus was able to maintain his position, unlike Caesar, is a bit more complex. There certainly remains an element of familiarity playing a part - after the years of Caesar's rule, then the triumvirate, the full republican system was a bit of a hazy memory - but there is more going on. One way of getting some insight into Roman attitudes is to look at Caesar's death. It is highly unlikely that it is purely coincidental that Caesar was killed shortly after being appointed dictator for life, cementing the idea that he would always be the sole ruler. Dictator was an official position within the Republic, but one reserved solely for emergency situations, and with a strictly limited term. The usual controls over consuls did not apply to a dictator, who on a literal lingusitic level said what would be done, without the possibility for debate or disagreement. By claiming the title for life, Caesar openly stated that he was acting outside senatorial control.
Augustus never claimed such titles. His official role was of princeps, the title traditionally granted to the most senior member of the Senate; the honorific Augustus was itself a senatorial honour. Instead of emphasising his superiority over the Senate, Augustus at least made it look like republican ideals were still being maintained.
The modern distinction you mention above is therefore a bit misleading. Caesar was never officially appointed king, it just seemed as though that was his aim. Similarly, Augustus never had a title which had the same meaning as the English 'emperor'. It comes from the Latin imperator, which is literally a general, 'one who gives the orders'; again, a traditional title taken directly from the Republic.