Basically Harvard, and maybe Columbia and that's it. UIUC, Wisconsin-Madison, Berkeley, Michigan-Ann Arbor are orders better schools in the sciences than Dartmouth or Brown.
Once you're at that level, the differences are marginal. It matters if you want to go to graduate school, I suppose, but we're talking about people who just want to get a degree and go into the workforce here. For them, Princeton vs. Michigan won't matter a bit if they're studying to become an engineer.
If you want to study applied math and go into the workforce, then my point still stands, going to Princeton vs. Michigan simply doesn't matter. If you want to go to graduate school, it's unclear if it matters all that much. Princeton tends to draw the most talented young mathematicians in the world, and has for decades. Presumably if a student of that caliber went to Michigan, they would do just as well, because Michigan also has talented mathematicians who do good research and can write good letters of recommendation for graduate school.
Specialty fields yes. But Princeton engineering is ordinary, and the state schools often have exceptional faculty across multiple departments - leading to way more collaborative, cutting-edge work.
The undergraduate students at Dartmouth and Brown are generally much better than the ones at those large publics. The rankings are largely due to the department size and what the professors and their PhD students are publishing.
