It depends what you want to do with your Math education. Both are challenging, rigorous schools yet they have very different strengths of their undergraduate programs.
If you’re interested in the research route than Dartmouth is better than Penn. Dartmouth, being undergraduate focused, would provide you more opportunities to work on high-level research problems, immerse yourself in university mathematics, and build close connections with Dartmouth’s math department. Dartmouth’s non-professional graduate programs tend to be pretty unremarkable, but they have stellar graduate school placement (a function of having close connections to professors and an exorbitant number of opportunities to do research). Penn’s math department is much better known on the graduate level, yet it isn’t spectacular either. If you intend to go to graduate school (for mathematics or any other subject), Dartmouth would be a better intellectual environment than Penn. Penn’s undergraduate math department would still be great, but the size of the university makes it difficult to become close with professors, get involved with mathematics research, and find mentorship for your own work.
Now, if you wanted to go into finance, either quantitative finance or some flavor of banking, both Dartmouth and Penn have their advantages. Penn has Wharton which is THE school for undergraduate business. Since you’d be an applied math / pure math major, you would not be in Wharton, although you there are ways to co-enroll between the engineering school or the college of arts and sciences with Wharton. Wharton will definitely open more doors in the finance world, but the differences in amount of job opportunities on Wall Street between Penn and Dartmouth is not significant. Dartmouth is more of a liberal arts college, hence it doesn’t have the same pre-professionalism that Penn exudes, yet Dartmouth has an incredibly strong alumni network on the East Coast. While Dartmouth is not as well known as Penn (and Wharton), Dartmouth’s alumni network on Wall Street tends to heavily favor Dartmouth alumni. According to this website http://www.bestcollegevalues.org/top-alumni-networks/, Dartmouth has the fourth strongest alumni network (and ~40% of Dartmouth graduates are in management positions). Additionally, Dartmouth has a 49% alumni giving rate (compared to Penn’s 8% rate), and according to 538, has one of the best return on investments for any college. At one point, Dartmouth had the highest mid-career salary out of any school in the country (you can find the Forbes article - interestingly enough this study happened prior to 2008 when Wall Street / American Financial Sector was still the most profitable industry). Another huge advantage of Dartmouth is its D-Plan. The D-Plan allows you to take terms off and study abroad, work on research projects, intern at companies, travel and research, etc.
If you want to go into business (Jeffrey Immelt, the CEO of General Electric, was a Dartmouth Applied Mathematics major), both Penn and Dartmouth will give you about equal opportunities. Penn’s advantages in business are its urban environment, the Wharton School, its greater reputation (this is tricky - a lot of people haven’t heard of Dartmouth while most people think “Penn” refers to Penn State which is why most Whartonites say they go to “Wharton” and not “Penn”), and its pre-professional culture (which might not be so appealing for some people). Dartmouth’s advantages include its super tight-knit and loyal alumni network, undergraduate focus, and graduate school placement.
Both schools have their idiosyncracies, and they are both referred to as the “Party Schools of the Ivy League.” Dartmouth and Penn are both fraternity-centered, although Dartmouth’s frats are very inclusive and welcoming. The Dartmouth community is much homier than Penn’s, but Penn is located in a nice part of Philadelphia with strong ties to the city. Other than Greek Life at Dartmouth, skiing, hiking, outdoorsy activities, tree-hugging, etc. are popular ways to spend time. At Penn, you have nightclubs, bars, restaurants, concerts, plays, etc. Philadelphia isn’t New York, but the City of Brotherly Love still offers much more than Hanover.