Dartmouth offers a few advantages over Williams in fields outside of academics. In terms of academic reputation, Williams enjoys a few distinct (and I think, a little overinflated) prestige boosters that make it seem much better than its peer LACs and Dartmouth.
When thinking strictly about academia, Williams tends to have more cachet over Dartmouth (as Dartmouth’s intellectual reputation was soured by a series of sensational articles over the past few years). Given how influential college rankings are in shaping public opinion, the fact that Williams is at the top of LAC lists and at the top of Forbes (I don’t know if it’s still #1 - but Forbes has a terrible ranking system; how can you rank Stanford and MIT on the same list as Pomona and Williams?) does carry some weight. This isn’t a problem (as I will later explain), but I understand that when you’re applying for college, these “rankings” can shape you opinion.
Now all of these factors only talk about “academic reputation;” they have nothing to do with academic quality, grad school placement, and job placement. Williams has cultivated a reputation as one of the most intellectual schools in the country. Yet, reputation doesn’t accurately capture the quality of the student body and academic achievement. Speaking anecdotally, the friends I have at Williams are very intelligent, but their peers are a mixed bag. Some of them are highly intelligent and some of them are just your run-of-the-mill jocks who come from East Coast prep schools. But this is to be expected with any of these schools. Williams has a huge sports community that serves as a kind of replacement for the lack of a fraternity system (meaning the athletic houses serve as replacements for Greek houses). Dartmouth has a high proportion of athletes as well (and does surprisingly well in many Ivy League sports), although they also “enjoy” (for better or worse) a large Greek community (which is changing as of now).
In terms of overall reputation, Dartmouth has a much stronger reputation than Williams especially within business circles. Dartmouth and Tuck have greater global influence, a larger alumni network, larger financial resources, and a strong community. This is not to say that Williams is lacking; Williams has a “mafia-like” alumni network with intense devotion to the school. But, as I explained, Williams has a stronger reputation in terms of academics. It is seen as the more intellectual school (even though they are very similar in terms of the social scene - one has a Greek system and the other has a clubby athletic community).
Now to the more relevant parts of your question (I hope the above was helpful, but I know it’s kind of unrelated): Williams is smaller than Dartmouth which could have its research advantages (Williams undergrad is about half of Dartmouth, but Dartmouth has over twice the resources that Williams does so it isn’t really a big issue). In terms of graduate school placement, Williams grads pursue PhD’s much more frequently than Dartmouth grads, but Dartmouth has one of the highest business, law, and medical school acceptance rates in the country. Williams also has a very high rate of b-school, law school, and medical school placement as well; both schools will provide you with the opportunities to get into your ideal grad school.
In terms of the math program, it’s also a toss-up. Williams attracts more students focused on “research mathematics” while Dartmouth attracts more students looking to go into business and engineering (Williams doesn’t have an engineering school but they do have a program with Dartmouth where you spend 3 years at Williams getting a degree and 2 at Dartmouth getting an Engineering degree; they also do this with Columbia). Both schools have dedicated departments looking to give students tons of research opportunities and internships. Dartmouth does have a wider spread of “applied mathematics” topics (they have one of the most underrated CS departments as well as programs in financial mathematics, engineering mathematics, operations research, etc.). I’d investigate each school’s math department page for more information about course offerings, research opportunities, and faculty focus.
In terms of job placement, both schools will serve you well. Same with Amherst, Swarthmore, Middlebury, etc. Dartmouth has the bigger network, more entrepreneurial resources, more financial resources, and more managers. Dartmouth graduates also have some of the highest mid-career salaries. Williams has an equally tight-knit following. As long as you are a good student and involved in your community, both schools will prepare you for great job opportunities.
Essentially, what it comes down to is the community. Dartmouth’s curriculum and school is more unique than any other similarly esteemed institution (with the D-plan / quarter system, the tight-knit alumni network, the outstanding business school, the undergraduate focus, and the Greek life). Williams is much more intellectually driven (at least, on the surface level - my feeling is that the two are much more similar than the rankings will let us believe), smaller (hence greater interaction between people), and less preprofessional. Both schools have high percentages of prep school kids, although Dartmouth has historically been more diverse.
In all, both are great schools. Dartmouth has had a bit of an image problem recently, which Hanlon is working on improving. Williams has always been highly esteemed among those who have heard of the school. The math programs are both solid, and I don’t think finding employment will be a problem for you (even though I don’t know you, the fact that you’re thinking about it now suggests that you are likely to work hard in college). Grad school placement is kind of split, as Williams has more “research-oriented” kids while Dartmouth has the business folks. Hope this was helpful.