I think much of what Oberyn said about Williams is true, but I’d offer a few corrections. First, Dartmouth is not really more diverse than Williams – Williams’ overall diversity has been trending up in recent years, and in the class of 2019, over 50 percent of the student body consists of international students or American students of color (the rate for the entire student body is more like 45 percent or so, but that will soon be over 50). Williams is also a bit more economically diverse than Dartmouth (although, frankly, all of these schools are overwhelmingly composed of students from the upper middle class or above on the economic spectrum), with one of the highest percentages of Pell Grant recipients and first-generation students among elite schools. Second, it’s unfair to say Williams is dominated socially by sports teams – the majority of the student body are not a member of any sports team (I was not, myself, and had a vibrant social experience), and unlike at D1 schools like the others on your list, the athletes on the sports teams are typically not defined by their sport – most of them are engaged in other activities on campus as well, not just athletics. Williams is certainly a sporty place, with a lot of athletic or outdoorsy students, but I got the same vibe from Dartmouth (as well as virtually every NESCAC school, the exceptions being Wesleyan and Conn College).
What I love most about Williams is that the students are just as smart as they are at the Ivies or Stanford, but (with some exceptions of course) lack a bit of the arrogance and sense of entitlement you’ll find at more universally well-known universities. This is also I’m sure an overgeneralization, but I was really turned off by the students I met at a few of the Ivies (in particular Princeton and Harvard, less-so at Dartmouth; I’ve had little exposure to Stanford). I do think Dartmouth and Williams have many similarities; I didn’t apply to Dartmouth however because of the dominance of the frat scene, which was something I had zero interest in. It also felt like a more conservative (along several dimensions, not just political) place to me than Williams. Williams’ social life was in no way exclusive or hierarchical, and really everyone seemed to know each other and get along fairly well instead of being segregated into narrowly-defined groups, which I think is one of the best things about the school. People have their groups of close friends of course, but you can also float freely through many different social environments. On the downside, by the time you graduate, it feels like you know (at least by reputation) nearly everyone in your graduating class, so to some that can feel a bit cloistered. But it also builds a tremendous sense of community.
Finally, I’ll give you a short list of some of the more idiosyncratic things that I feel make Williams truly unique among its peers, beyond just stellar academics and a beautiful setting; you can read up on any of these on your own. Some of these are more important than others in campus life, but in combination they give some sense of the unique vibe / feel that distinguishes Williams among its liberal arts peers:
The JA and entry system
Mountain Day
Tutorials
Winter Study
The purple cow mascot
Semester’s-end Trivia
The WCMA walls program and the popularity of Art History, generally
Chapin Rare Books library
The bordering-on-obsessive interest in the school among many of its alumni
The Amherst rivalry (few more long-standing or fun or natural college rivalries in the country)