Mary Beard stated that the assassinations of Julius Caesar and Caligula reflected the shift in Roman Politics, explaining that when Caesar was killed none of the conspirators attempted to murder his remaining family, while in Caligula's circumstance the assassins murdered his whole family (except for Claudius). My question is why? Why didn't the assassins kill all of Caesar's family (his wife etc.) or why did the assassins kill Caligula's family (wife etc.)?
As far as I know Claudius wasn't killed because most saw him as a bumbling fool. He had a speech impediment and was quite slow, or at least seemed that way.
But the most important difference here is the form of government at the time. While Julius Caesar had ambitions to become the sole ruler of Rome, he was not emperor. His position was not hereditary. In practice of course Augustus took the throne after Caesar but he pretended that Rome was still very much a republic. While today we call him emperor, he himself did not. So the reasoning behind not killing Caesar's entire family was that with Caesar dead the republic had been saved.
Caligula on the other hand was an emperor. His position would be inherited by a family member. To get the one you want on the throne, you'll need to kill those you don't want on it. As I said Claudius wasn't taken very seriously and people felt he could be easily manipulated. This made him the best man for the job as he had a claim to the throne.
So the shift reflected in these murders is that from a republic to an empire with a hereditary leadership.