They haven’t really shown a preference to either one since the cost for them would be fairly similar for each school (I’m sorry if I’m making the financials confusing but the only true difference in cost would be on my end).
My vote then is for RPI. It will be a gift to come out of a school debt free.
How soon can you start doing research at RPI? Will you be able to skip intro Physics classes due to AP/IB credits and start taking upper level classes sooner? A big part of getting into a Physics PhD program(apart from the obvious high GPA) will be research and the Letters of Recommendations you can get from professors. Many schools will expect you to be at a certain level classwise before you can get into research so the sooner you get there the better and the more time to cultivate those relationships for your LORs.
At RPI I would be skipping the first year math classes but they HIGHLY recommend I don’t skip the first year physics classes since they want all physics majors to take their Honors Physics I & II, so even though I could skip due to AP credit I wouldn’t be.
At Manchester they do still go over the entry level physics stuff but they do move on a bit too since part of my acceptance to Manchester was dependent on my AP Physics C scores (I needed a 5 on each one).
If you don’t have 60K or a significant portion of that saved up, choose RPI. 60K is quite a bit of debt to graduate with and you could be hard pressed to find loans going up to that amount by yourself anyway.
RPI should be able to get you where you want to go if you do your share toward getting there.
If you have 60K - or even half of that - saved up and it can come out of pocket, then choose the place you like.
Trust you worked out living costs in the UK to add into that cost.
AP physics C is not as math intensive as college physics courses for physics majors are, particularly in E&M. Taking the honors intro sequence could be a good option.
Are you a US student who can file the FAFSA? If so, you can borrow a total of $27k in federak student loans ($5,500 freshman year, $6,500 sophomore year, $7,500 junior year, and $7,500 senior year). Any debt beyond that would require that you find a co-signer or someone to borrow the money for you.
It’s truly bizarre that a difference of $5K per year in net cost would mean you paying $60K more and your parents $40K less. The actual difference is only $20K and that assumes you do 4 years in the UK to get a masters. Doing a BSc in the UK would actually cost less overall than the equivalent US degree.
I can see how it’s weird so let me try and explain a bit: RPI’s ticket price with room and board, tuition, etc (the total cost) is about $75k a year so I would need to pay about $25k based on the plan I have with my parents. However, RPI offered me a scholarship (the leadership scholarship they called it) which is ~$35k. My parents said that any scholarships I got go towards my third so RPI would cost me very little due to the scholarship. Manchester’s ticket price (again, with room and board and everything) is about $45k but I am getting no scholarships so my third is $15k a year ($60k for 4 years as the program I’m in is their integrated masters). Thus, even though the total after scholarship price for RPI is only about $10k/year or so less than Manchester my price is drastically different.
I hope that clears the waters a bit.
I will say I haven’t sat down with my parents and worked out all the details yet as AP scores only came out yesterday so my ability to attend Manchester is a recent development. The cost on my end could end up being less than $60k once I talk to them but right now I’m stating prices based on the plan we had previously agreed upon.
Looks like your parents are effectively forcing you to choose RPI over Manchester based on how they set up the finances.
Every family has their own way of putting together a plan, but it sounds like you are saying your parents are offering to pay 2/3 of whichever school you choose, no matter how expensive, and that any scholarships you get will go toward your 1/3. If you get into a higher priced school, but receive a scholarship, they are willing to pay much more in order to send you there. But, if you get into a lower priced school, they will draw the line at paying much less (2/3 of the original COA in both cases.)
If I understand correctly: hypothetically, this plan would essentially force you to choose a school that offers you a scholarship, even if the school is much more expensive and significantly less prestigious. (I’m not saying this is the case with your particular schools, I’m just describing the plan.) Ironically, your parents would then end up paying an amount that is GREATER than the full cost of the less expensive, more prestigious school. Do I have this correct… that their objective is to require you to pay 1/3 of anything, minus scholarships, even under the circumstances I’ve described?
I guess that’s one way of doing things, but I wonder if they would see their idea as a good one if it was laid out this way. So many parents would be thrilled if their own total expense was less!
I am NOT saying that I know anything about these two schools or that the difference in prestige, quality of education, or any other factor is particularly significant. I’m just pointing out that your parents may not realize that there are two ways that you as a student can contribute to a lower COA… one is scholarships; another is choosing a school that costs much less.
As a parent, if you willingly turned down the higher-priced school for a lower-priced one, I would consider that to be similar to taking responsibility for 1/3 COA at the more expensive school. I would see that you lowered my contribution by MORE than if you chose the more expensive school. Personally, if I was willing to pay 2/3 of the COA at the more expensive school, and the entire COA at the less expensive school cost less than that, I would pay the whole thing and send you there without you needing to incur the debt. I get what your parents are trying to teach you in terms of fiscal responsibility, etc., but maybe it would help to frame it for them in this way.
Ironically, you actually WANT to go to the less expensive school. So many times, it’s the other way around. I hope your parents will reconsider their plan, and appreciate that, scholarship or no, you would actually be significantly cutting family expenses (yours and theirs put together) and flat-out saving them money by choosing school #2.
Like I wrote, I think I understand your parents’ financial plan and the comparative cost of the two schools in question. It goes without saying that if I’ve gotten all of this wrong, please go ahead and ignore the post. 
Lots of questions: Hope you ask this one. Are you completely certain that EVERYONE in your chosen field will know that UofM is better? Will they accept a British degree and hold it to the same value. You may know that UofM is better but does everyone else.
You could also split the difference. Go to RPI undergrad and apply to UofM for the Phd. Makes it free and if you think UofM is best, then you’ll gain in both respects.
Honestly I don’t think the UofM degree is worth the additional 60K ( and I normally lean towards more debt for a better degree unlike some posters who always say no to debt).
The ‘prestige difference’ between U of M & RPI is simply not worth $60K debt. Even though the PhD will be covered, you will still spend many, many years paying it back, and it will inevitably shape your choices as you come out from grad school.
I’m assuming you used the UN News global ranking in physics, where Manchester is 43 and RPI, 523! I don’t know how accurate that is, I grew up in upstate NY and RPI had a pretty good reputation for its physics dept, but that was in the 80s, so things may have changed. If those are even somewhat accurate, that’s a huge difference and I think may put Manchester back in play as the choice. Also don’t worry about the MPhys vs BS, most, if not all, PHd programs in the US will give you a masters as well if you want it, along with your PHd. A friend was applying for physics PHd at Cornell and he said they don’t have a masters, you go straight from BS to PHd. Good luck!
I don’t know anything about either of those schools, so I can’t comment on the level of prestige. However, as someone outside of the physics realm, I don’t think the University of Manchester is that well-known in the US (not like Oxford or something).
Are you 100% sure you’re going to do a PhD? Would you still think it’s worth it if you change majors? What if you change to something that requires a Master’s degree instead? The PhD factor shouldn’t change how much debt you’re willing to take on, IMO – in fact, you’d likely make less from your PhD stipend than you would working with a bachelor’s degree, so you’d have to wait until you get your PhD to start paying off your debt.
And isn’t the conventional wisdom that prestige matters LESS for undergraduate programs if you’re planning on graduate school? Employers won’t care about how prestigious of a college you went to as an undergrad, just where you went for your PhD…maybe focus on prestige then. 60K is a lot, although it’s doable.
“You could also split the difference. Go to RPI undergrad and apply to UofM for the Phd. Makes it free and if you think UofM is best, then you’ll gain in both respects.”
That is not likely to be a feasible option. PhDs are not funded by default in the UK, you need to get a scholarship or research grant to cover the costs or else you pay yourself. Most grant funding that the universities have is for UK/EU students not internationals - so you may end up limited to seeking ultra-competitive scholarships (eg Marshall or Fulbright scholarships).
That’s because you just do research and finish in 3 years, you aren’t TAing and taking 6 years as is more typical in the US.
@theloniusmonk I didn’t look at USNWR - but if you look at QS you’ll find RPI in the 150-200 tier 
But more importantly, those rankings don’t speak to the undergraduate physics program in particular- and those are (comparatively) flat, especially in the first 2/3 years. My physics kid, and her friends in Ireland, Scotland, and Belgium compared their undergrad physics degrees (they are now all in grad programs), and the differences between what the European kids did and what my US-based LAC kid did physics-wise was considerably less than any of us imagined it would be (especially as they did physics 100% of the time for 3 years, where mine did it 60% of the time for 4).
To be clear, UM is a great university, and in a world where the OP could afford to go without taking on the debt of a new sports car I would be all for it. But the specific question was, how much is the extra prestige worth? and imo, it’s not worth the debt. Prestige of undergraduate school is not a big factor in physics grad school admissions, and the OP can absolutely get into any physics grad program from RPI.
Does the prestige comparison matter at this point? The OP’s parents are financially forcing him/her to attend RPI (despite RPI costing them more). It does look like RPI graduates in physics do commonly go on to graduate school, but the OP may want to contact the department directly to find out how many years and total graduates the listed examples of prestigious PhD program matriculations come from.
https://science.rpi.edu/physics/programs/undergrad/bs-physics
I am not a fan of comparing the “prestige” of different universities. Instead please focus on the resources and opportunities available to undergraduates.
I am a physics professor who is familiar with the physics department at RPI. Robert Resnick (of Halliday & Resnick fame) and his successors developed a strong introductory physics curriculum there. The department offers all of the courses and research opportunities necessary to prepare for physics graduate school. RPI is well known to US graduate programs and their alumni do fine in graduate school.
I doubt that the University of Manchester would give you a leg up over RPI in US graduate program admissions. As others have noted grades and GRE scores count, but research experience and recommendations from professors are the most important factors. RPI can offer you those things.