I do not state that a particular university is better than another university. One or the other may be a better fit for you. I can provide some specific information for you.
Williams has 11 faculty in computer science and Princeton has 63.
Williams has 28 math faculty and Princeton has 98 math faculty and 16 in applied math (ORFE)
So Princeton can offer more breath of courses and you will have a greater opportunity to find a prof who is an expert in a field that you would like to conduct independent research.
Princeton and Dartmouth are generally considered to be the Ivies that have the most school spirit. Princeton alumni have one of the highest contribution rates in the entire country; so I guess they were happy.
Most Princeton students have an international experience. The endowment at Williams is $2.75 billion. The endowment at Princeton University is $25.9 billion. The substantial endowment at Princeton enables the university to provide international trips to students. Students who have completed the HUM Sequence course are invited to take a trip to Greece or Rome during fall break of their sophomore year; costs for the trip are covered by the University. A recent Shakespeare course took students to Stratford-upon-Avon so students could see a Shakespeare play in a one room theatre with the audience on three sides. English majors are eligible for a fellowship to study English literature at Lincoln College at Oxford University. Recent courses have taken students to Havana, Peru, Mexico, Japan, Kenya, Bermuda, Panama, France, China, Brazil, Spain, Portugal, Russia, Argentina, and probably many more that I have missed.
The faculty student ratio at the two universities is similar. Princeton is larger with 1,238 academic staff and Williams has 313 academic staff.
Whether or not Princeton is pretentious is a decision you can make best by attending Princeton Preview days. I believe that you will find the students similar to yourself. I am amazed how much misinformation there is on CC on Princeton’s eating clubs. One third of the students choose to continue having their meals in their residential college. One third join a club by telling the club that they want to join that club. One third join a club through a process similar to fraternity rush. Many students join a specific eating club because they have made friends with members of that club through a student organization. Students are not allowed to join a frat or sorority until their sophomore year. Greek life is not important. The Greeks do not have “houses.” Student do not take their meals in the eating clubs until their junior year; this reduces their impact compared to Greeks at other universities.
Many consider Princeton’s math department to be the number one math department in the world. Alan Turing, called the “father of computer science” received his PhD from Princeton in 1938. John von Neumann joined the faculty in 1930. All computers today are designed with a single memory for programs and data first described by von Neumann in a paper in 1945; computer scientists call this the von Neumann architecture. The first stored program computer, the MANIAC (mathematical and numerical integrator and computer) was built in Princeton starting in 1947. The first commercially available IBM computer, the 701 was built based on the MANIC design. Princeton engineers have had an important role in the development of networking and the Internet. David Boggs BSE ’ 72 was a co-inventor of Ethernet. Robert Kahn MS *62, PhD 64 co-invented the Internet protocol TCP/IP. The largest cloud service provider with over 2 million servers was started by Jeff Bezos EE & CS, ’86. Stanford, Princeton, and MIT are frequently cited as having the best computer science PhD programs. Google has established an AI lab across the street from Princeton University. The lab builds on several years of close collaboration between Google and Professors of CS Elad Hazan *2006 and Yoram Singer, who will split their time working for Google and Princeton. Eric Schmidt BSEE ’76 is the former Executive Chairman of Google. The engineering school founded in 1889 was one of the first engineering schools in the US.