Octavia was the last relative of Augustus [EDIT: Julio-Claudians are confusing! I think this is right now?]. Nero only got in the family by adoption. He had Octavia killed after he divorced her; in the drama Octavia there were riots in Rome to restore her marriage - it was her connection to Augustus that gave Nero legitimacy. Nero killed her, preventing the possibility of forming another such connection.

Claudius also did not have a direct connection to Augustus - his ancestor was Augustus' sister Octavia, and that might have had some influence on his decision to marry Agrippina (who did have a direct connection) over Lollia Paulina.