What's really interesting is that it's not valid to simply knee-jerk to thinking that we're seeing the whole thing repeating, just look at the book, and we all know how it ends.
One of the biggest deals I see that are similar are these two aspects:
1) when still a republic, the plebians (non-patricians) were increasingly open to emotional appeal from demagogs.
2) there was a gradually erosion of civic duty that was replaced by either slaves or hirelings. An example was that there was no longer an army composed of Roman citizens, but paid mercenaries.
The Romans were much more heirachically-based so far as social classes.
An interesting feature was how voting worked. Some people had more votes than others:
https://en.wikipedia.org/wiki/Social_class_in_ancient_Rome#Property-based_classes
So far as the encroachment of decadence (both the early Roman and early US citizenry were fairly puritanical and upstanding), it is probably a symptom more than a cause.
My opinion, only.
The complete 6 volumes runs to about 4k-4.5k pages with normal annotations.