One difference that might not have been a factor a generation ago, but which is definitely a difference today, is geographic diversity.
There’s been a lot of concern about non-resident enrollment at UC campuses, especially Berkeley and UCLA. But even so, more than 3 out of 4 Berkeley students are Californians, and since the UC system recently capped non-resident enrollment, that’s not going to change. Berkeley regularly turns down better-qualified out-of-staters in favor of less-qualified state residents (and as a state school, this is probably the right thing to do).
But USC (as a private school) doesn’t have to do that – they will cheerfully accept the highly-qualified non-residents that Berkeley doesn’t have room for (USC is also happy to offer them financial aid). The days when USC was overwhelmingly Californian are long gone. The Class of 2021 was only 42% in-state, with 43% from other states and 15% international. The in-state share will probably continue to drop in the future; other top privates in California, like Pomona, Caltech, or Stanford, only have around 25-35% in-state enrollment.
So USC is becoming nationally and internationally diverse in a way that Berkeley cannot duplicate. On the other hand, it’s probably true that there is more socioeconomic diversity at Berkeley.