The difference is that Caesar held positions within the Republic while Octavian/Augustus created a new position above it.

Let me quote an answer I gave yesterday to the question was Julius Caesar an emperor? for some more details:

The first Roman emperor was Augustus, Caesar's adoptive son. Augustus was born as Gaius Octavius, though in the English world we know him as 'Octavian'. Caesar adopted Octavian as his son in his will. After Caesar was killed, Octavian spent two decades first fighting Caesar's enemies and then his own enemies. Having emerged as the sole victor after decades of civil war, Octavian was granted the titles of Augustus ("Illustrous One") and Princeps ("First") in 27 B.C. As a victorious general, he also could use the title Imperator, which is the word where the English word 'emperor' is derived from. It are these titles which were (mostly) used by all emperors following Augustus. This is why we consider Augustus to be the first emperor.

Caesar did not have these titles; Caesar did not even ask for these titles. When Octavian was granted the titles Augustus and Princips his position was a lot more secure than Caesar's position ever was. Octavian maintainted the pretence of the Republic while subverting it with his new titles, but Caesar could not be that bold. All Caesar could do was usurp existing mechanics in the Republic. Caesar served as consul (the highest regular office in the Republic) several times in a row, even though that officially wasn't allowed. Near the end of his life Caesar secured to position of dictator, an extra-ordinary position giving someone sole rule over Rome, traditionally for only 6 months. Caesar though was granted this position first for two five-year periods and then for life. At this point his enemies decided he had become too dangerous for the Republic, so they killed him.

So this is why Caesar was not an emperor: Caesar abused existing positions within the Republic (as others had done before him, perhaps not to the same extent). Octavian/Augustus created a new position with new titles, effectively ending the Republic and starting an empire.

To correct one misconception first, Augustus did call himself imperator. While the Senate gave him the title of 'Augustus' and 'Princeps' he also styled himself 'Imperator Caesar Divi Filius'. The title of imperator will from now on be reserved for Roman Emperors, and being declared imperator by the troops would become an automatic act of rebellion; this is how most civil wars in the Empire will start.

As for Caesar; Yes, he held the office of dictator for life, but he wasn't the first. Sulla was ratified as a dictator by the assembly in 82 BC with no time limit (and famously resigned it in the end). Only five years passed from Caesar the crossing of the Rubicon to his assassination, and a large part of it was spent fighting a civil war, so he hardly had the time (or the inclination, really) to reform the Republic into the Principate; it will be done during Augustus' 40+ year reign.

There are indications that Caesar might have wanted to be crowned king. His public rejection of crowning after people that were present reacted negatively, and the story that was distributed by Caesar's people which said that the Sybilline books said that the Romans could only defeat the Parthians if they led by a king. He never could state this publicly, of course, as Romans really hated the idea of being ruled by kings at the time, and it's a big reason why Augustus kept the necessary fictions of being the first citizen.

Back to Augustus, most of the political and military structure that ended up being the Principate were put in place by him: establishment of the Praetorian guard, the number and placement of the legions, pretense of upholding Republican offices such as the consulate, the division of imperial and senatorial provinces (while securing supreme authority of the princeps in nominally senatorial provinces), etc. During his reign, he did his best to maintain the pretense of the Republican institutions in order to distract from the fact that Rome was now a kingdom by a different name. By the time he died, hardly anyone was alive who even remembered the Republic.

Future emperors (at least during the Principate) will derive their legitimacy from princeps being a 'father of the people', and a Republican charade will be kept up through consulships, even though they will essentially be relegated to a ceremonial title. In practice the loyalty of the legions was the source of real power.

Edit: coherence

This too bugged me for years. There are two answers...

Both Caesar and Augustus where autocratic rulers of the Roman State, effectively wielding extremely similar sets of powers. Caesar did so within the framework of the republic. While August created a new power sharing agreement with Senate that transferred military power from the Dictators/Consoles to himself.

The other, and probably more important distinction, is that Caesars heir had to fight a civil far to cement his position. Augustus's did not, and his powers transferred down to the next generation.

These distinctions are kind of arbitrary. The transition from republic to empire took a long time, and Julius Caesar was one of several figures who seized more power than had traditionally been allowed to a single person. So why does Augustus get singled out as the turning point? Many reasons, but especially:

Unlike Marius, Sulla, Cataline, Gaius Julius, etc, Augustus kept power until his death, and power remained centralized in the hands of his successors (even when those successors soon started knocking one another off). After Julius Caesar's assassination, power went back to the senate and the new triumvirate; after Augustus, power remained in the hands of an emperor.

Augustus radically transformed the way the empire was administered. He reformed the army, he consolidated economic and political power to his new office that had formerly been decentralized in the hands of various strong men and members of the senatorial aristocracy.

And Augustus redefined old offices in new ways. He placed himself more firmly in the center of state religious practice than Julius Caesar had been able to do, he revived family values, he sponsored the writing of Roman histories and myths (Livy and Vergil), and he created new titles that simultaneously emphasized his commitment to old Roman values while subtly transforming them to focus themselves around the office of an emperor. Julius Caesar was a dictator for life; Augustus was 'first,' the patron of the empire who held all the strings. This allowed him to rise above the senate in a way Gaius Julius hadn't been able to do, and the senate never regained its social influence after Augustus' reign.

So Gaius Julius definitely helped kill the old system, but he wasn't the only person pulling it apart (just one of the most nearly successful). But Augustus did more than put the last bullet in the head of the Republic: he built a new system that restructured it. So even though he kept the cognomen Caesar that he received from his adoptive father, he built something different than Julius had managed to create that was, for the next two centuries at least, very new and successful.

Although modern historians do not consider Caesar to be the first emperor, the Romans themselves did.

e.g. Suetonius' book The Twelve Emperors, written about 121 AD starts with Julius Caesar as the first emperor.