I know this is a very broad question, and obviously their treatment of women will likely seem very antiquated in a modern view. I always thought that Roman society viewed women as people who were just there to have and raise children properly, that they didn't have much realistic influence.
However, I was reading just reading about Claudia Octavia, Nero's wife -- how Romans rose up in protest against her exile because of how beloved she was in their view, which makes me think my previous understanding of Roman culture was likely off.
So I guess I really have two questions;
Just finished writing my dissertation on Nero, with a focus specifically on the year 62, which is exactly around the time Octavia was exiled as you know. So I'll give a go at answering the second part of your question.
Tacitus is our best source here, and he writes about how the citizens of Rome were upset over Octavia's departure. She was the daughter of the previous Emperor, Claudius, and in his eyes a far better woman than her replacement Poppaea. A discussion of his presentation of both these women could take up a whole essay! Effectively however Tacitus talks of the citizens being so angry because it allows him to portray Octavia, the good and moral woman, as being removed unjustly by Poppaea, the terrible and morally corrupt woman, and reflect this decision back onto Nero.
If we believe Tacitus' accounts, the citizens are appalled at this, hence why they are so upset. In their eyes, the previous emperor's daughter is being replaced by a worse woman, and their emperor is insulting his position. Hence why they tear down Poppaea's statue.
I would highly recommend Ginsburg on this topic.