Julia was accused of an intrigue with Seneca, and was first driven into exile and then put to death.

Seneca was banished to the barren and pestilential shores of the island of Corsica.
Seneca, as one of the most enlightened men of his age, should have aimed at a character which would have been above the possibility of suspicion: but we must remember that charges such as those which were brought against him were the easiest of all to make, and the most impossible to refute.

When we consider who were Seneca's accusers, we are not forced to believe his guilt; his character was indeed deplorably weak, and the laxity of the age in such matters was fearfully demoralising; but there are sufficient circumstances in his favour to justify us in returning a verdict of "Not guilty." Unless we attach an unfair importance to the bitter calumny of his open enemies, we may consider that the general tenor of his life has sufficient weight to exculpate him from an unsupported accusation.
Of Julia, Suetonius expressly says that the crime of which she was accused was uncertain, and that she was condemned unheard.

Seneca, on the other hand, was tried in the Senate and found guilty.

He tells us that it was not Claudius who flung him down, but rather that, when he was falling headlong, the Emperor supported him with the moderation of his divine hand; "he entreated the Senate on my behalf; he not only _gave_ me life, but even _begged_ it for me.