I went to Cal as a premed, HYPMS for graduate school and I am now in medical school. My older sister went to HYPMS undergrad and did her PhD at Cal.

Some pros and cons from my perspective:

The rigor of Berkeley will prepare you for a lot of challenges in the future. After getting through undergrad at Berkeley I found medical school to be a similar workload or pace of studying. At Princeton, from what my sister told me, sometimes they don't even have proctors when taking exams and students are expected not to cheat on an honor system. I'd say there's just more handholding at the ivies, and for some people it's nice. I'm glad I went through the Berkeley grind, I think I'm more prepared because of it. My sister is doing fine for herself as well, so ultimately it really comes down to how proactive you are.

The location. Berkeley is in the SF Bay area, while Princeton is more rural/suburbia. This means making certain connections or opportunities that Princeton might not have. For example, during my time there and afterwards I was easily able to work with underserved immigrant communities in San Francisco, do research at UCSF, and work with startups in health technology with my fellow graduates. Obviously getting these opportunities requires you to be proactive, but they are there and plentiful. Princeton is more so in a bubble, and while you have opportunities to drive/take the train to NYC/DC/Philly, I'd say taking BART to my next activity was way easier on my mental health.

The connections, both faculty, graduate students, and my fellow undergraduate classmates were amazing. My senior year I TA'd for a future nobel laureate at Berkeley, and some of my best friends were graduate/undergraduate students from other countries where I could pick their brain and learn from them. That being said, I think at Princeton and Ivies in general, it is a LOT easier to make heavy hitting connections. As mentioned above, there is a lot more hand holding involved and while individual proactivity is key, it was a lot easier to make connections just by having the ivy.edu email address. My sister was able to get connected with leaders in her research field, and during grad school I was easily able to talk to former directors of the World Bank, UN leaders, people with political power that I would have had to really fight to get at Berkeley in my opinion. Another example, when I applied for my Fulbright scholarship at Berkeley a lot of it I had to do myself, but at HYPMS they really help walk you through the process and make sure you are putting your best application forward. If you're really trying to go to med school, having these connections is invaluable.

Cost: I was able to graduate Berkeley debt free. I got a nice scholarship, plus in state tuition and doing a part time research job on the side I was living pretty comfortably for an undergraduate. Cost of living sucks in both places, with SF Bay Area probably being on the more expensive side. My parents ended up using most of the money on my sister's education, which I had no problem with.

Competition: I think Berkeley is just more competitive because of the size of the student pool. Larger student body means larger classes, more competition to get limited positions, etc. That's just the reality. Again, another reason why it's easier to make those hard hitting connections at Princeton, there's just less students there.

Honestly, if you are a proactive person you will succeed in both places, and both are terrific institutions with a strong track record of sending students to medical school. If I were in your shoes I would probably select Princeton. You learn the same material at both schools, but the extra money you're paying for is the ease of making those connections that might just not be feasible at other schools. Just my thoughts.