There are definitely some similarities–private research universities, very selective, Gothic campuses, and access to large city amenities are the most obvious.
In terms of differences, BC is a bit bigger with D1 sports. I do tend to think region can make a difference in terms of cultural norms/vibes.
BC has recently been relatively more white-domestic than WashU, WashU more Asian-domestic and Black-domestic than BC (although still a plurality white-domestic). I think it would be fair to say WashU is closer to the sort of “standard” ethnic mix for a very selective “national” college.
It appears from recent reports the effects of the Supreme Court decision have not changed this relative relationship much, although apparently the percentage of Black-domestic enrollees dropped at both, and a little more so at WashU closing the gap a bit.
Of course BC is Jesuit, WashU is secular. Student religious affiliation isn’t reported the same way, but apparently around 70% of BC students are Catholic (so more than, say, Georgetown, but less than Notre Dame). I’d suggest the above differences in ethnic mix are mostly an effect of that difference, given the typical ethnic mix of US Catholics (relatively high percentages of Hispanic and non-Hispanic whites, relatively low percentages of Black and Asian).
BC has no Greek life. WashU has some, but it definitely does not dominate the social scene, it is just an option.
In terms of curriculum, if the student would be in Arts & Sciences, I know a bit about that (less so Business or other schools). Consistent with the typical Jesuit approach, BC has a fairly extensive core requirement, albeit with some flexibility:
WashU A&S calls its gen ed requirements the Integrated Inquiry (IQ) Curriculum, and it is pretty complicated but the general idea is to allow a flexible, personalized curriculum plan that will include a range of fields and then also multiple areas of concentration (in fact it appeared to me lots of WashU kids end up with multiple majors/minors because that is one way of satisfying that concentration component):
https://artsci.washu.edu/finding-your-path-curriculum-philosophies
As usual, I don’t think there is an objectively better or worse here, but I do think these end up pretty different colleges overall. Not that this is the only way to think about these things, but in my circles, BC tends to be most popular with kids who want to be in Boston/New England/East Coast, want D1 sports, and (maybe) are Catholic. WashU A&S appeals to kids who see St Louis as an interesting city, are a little more focused on academics, and (maybe) find the exploratory/multidisciplinary approach at WashU appealing.
Edit: Oh, the other thing I think appeals to some kids about WashU is the South 40 experience, particularly for your first year. It is a dedicated area for underclass dorms with a lot of amenities, and the dorms pair up as residential communities for various purposes:
I assume you saw this on your visit so sorta forgot to mention it, but I do think it is among the factors that some kids in my circles consider when making a final decision.