Hello, I would like to state before I start that instead of Augustus will be referring to him as Octavian because it was an imperial name made for him when he was Roman Empower
With that out of the way, it comes down to many factors with the five main ones being:
I will go through each point one by one.
So, Octavian was named Caesar's heir in his will where he inherited Caesar's possessions. We cannot speculate as to why Caesar did this but our best guess comes from points two and three. This will gave Octavian immense political power when compared to Caesarion and basically overnight. Caesarion, by contrast, sure had the throne of Egypt due to being the son of Cleopatra but the wealth of Egypt hardly translates to that of Rome.
This leads us to the second point that Octavian was significantly older and of Roman Heritage while Caesarion was an infant and half-Egyptian. Octavian was 18 years old when Caesar died while Caesarion was roughly 3 or 4. Octavian was also a Roman citizen and was also from a Roman family. Caesarion by contrast was the son of Egyptian royalty so he was always viewed as a foreigner by the Romans. It was also because Caesarion was in Egypt with Cleopatra while Octavian was in Rome.
This brings another point that Caesar was quite fond of Octavian due to him being older and thus able to more closely connect with Caesar. Caesarion being a baby mostly under Caesar meant that he didn't know what legacy he would leave. That leads us to the last point of political power. Caesarion literally couldn't be active in roman politics and no one watching over him tried that much either. Octavian on the other hand made himself instrumental in roman politics by forming the second triumvirate.
Anyway, I hope that answers your question.
Seth,
(Edits are for grammar Mistakes and Corrections)