The Difference Between Mathematics and Applied Mathematics at Stony Brook
Here is my assessment of MAT and AMS (I double majored in both).
AMS is definitely more about getting a job. I really recommend doing AMS + compsci minor rather than AMS MAT double major if you care about a job and not about graduate school in pure math. Some of the students have gone on to become actuaries, the AMS department offers 2 courses that helps you pass the first 2 exams before you graduate (this is standard actually but a good head start nonetheless).
A lot of the professors are quite good, very experienced in their field. Basically if you are worried about your career opportunities, I really recommend an AMS/Compsci double major. It might take longer, but you have a lot of career opportunities since you know statistics and programming. Very very useful. They also have some good contacts, you just need to let them know your intentions early. Stop by Alan Tucker or Joseph Mitchell's office and have a chat with them, both are very nice guys.
The MAT route, many will disagree, is pretty much if you want to go to a pure math PhD program. I have not heard of too many MAT majors getting jobs out of undergrad, this is just my experience. But on the other hand, all the good MAT majors go on to graduate school.
Career wise, MAT is really about I want to become a research professor. AMS you can still go to graduate school but have some nice employment opportunities. AMS also lets you get a BS/Masters combined degree in a field like stats, operations research and something else that I forget. They also have a good mathematicial finance professor in James Glimm (who is also a world class pure mathematician) and he seems to still be active.
Just another word to the wise, for pure math majors, Linear Algebra is considered one of the most important courses. Now 211 is the numerical one, where all you do is compute, but 310 is really the first "pure math" course you take. It's all proofs and abstract reasoning. It's used almost everywhere in math and many math professors say if you don't understand or like linear algebra, you probably won't like pure math. With MAT, you'll get more of the same with 310. Everything will pretty much be proofs, be long winded and frustrating. You have to like doing proofs. That is the name of the game.
I would like to say that a reason a lot of MAT majors don't get jobs is probably because that's not their main goal; their main goal is to get into grad school and usually pure math grad school. Unless you really want to do pure math, I wouldn't major in MAT.
Now if you like problem solving AMS is a good option. Some of my AMS courses were actually more FUN than my MAT courses. If I had pursued AMS as heavily as MAT, I probably could've gotten a nice paying job out of undergrad.
So to summarize:
AMS Pros - good employability, good range of courses which prepare you for real jobs (statistics, probability, actuarial, operations research) and a double major with compsci is really a great idea. I say this because with any real applied math job, you need to know how to program, either statistics programs like SAS or with computer programming like C++ or Fortran.
AMS Cons - Hard to say, a con depends on context of who is asking. If you want to go to graduate school for pure math, AMS does not really help that much. Not proof intensive.
MAT Pros - If you do well, not just in class but with research and grad courses, you will get into a very good graduate school (If an idiot like me can get into top level grad programs, anyone can).
MAT Cons - Very little career training, but this should be understood from the beginning. You don't major in pure math to get a real life job. It happens, but applied math has so many more industry jobs. It's not even fair to compare them. And most industry jobs for pure math majors (i.e. I'm talking about those mathematical finance jobs on Wall Street) require a math PhD from a top school.
I liked almost every class I took in AMS and MAT, but I also love math, so again, context matters, it might be different for you.
Here are some potential careers coming out of an AMS with programming skills:
Statistician (rated the BEST job by the Bureau of Labor or the BLS)
Actuary (considered a top 5 job by the BLS)
Mathematical Finance (need to do a masters or a PhD, but if you do, it's big money. James Glimm who runs the math-finance department at SUNYSB has good connection with James Simons, the hedge fund mananger billionaire).
Operations Research Analayst
Code breaker at the NSA - the AMS department actually offers a course on code breaking, i think it's AMS 351. It teaches how you to encode messages using number theory and groups and how to break those codes sometimes.
These are all very nice high level jobs.
So the point is: AMS - better for your career, MAT - if you want to become a research mathematician. But of course if you do AMS, you can also go to an applied math grad program, very easily too. The faculty at both the AMS and MAT department are pretty friendly and will do their best to see you succeed. However, if you do just AMS, you will not be competitive enough to get into a pure math grad program.
In my opinion, both programs are really damn good. I loved almost every class I took in both majors. They both have some excellent professors who will dedicate time to their students. It's just they offer different things. In AMS 312 the mathematical statistics course, it was very hard and had lots of proofs. Then I had some courses where we didn't prove much but assumed a few key facts and solved some really cool problems, like in AMS 345 - Computational Geometry with Joseph Mitchell. We worked on robotics motion planning, where to place cameras to catch art thieves, very cool *****.
I would say the real "test" of whether or not you like MAT is when you take MAT 310, 320 and 313. These are the basic foundational courses in MAT. They are all proof intensive and very abstract. In 320 you rigorously prove all the facts you used in differential and integral calculus. 310 you take about more linear vector spaces, matrices, rigorous definition of the determinant and trace, complex operators, spectral theorem, etc.
Another way to test the waters, instead of talking to some math professors, spend an hour or two looking at the 310, 320 textbooks in the library. The linear algebra book is "Linear Algebra Done Right" by Axler and the 320 textbook is "Introduction to Real Analysis" by Bartle and Sherbert. Both are available on reserve at the Math Physics library. Really look through them. You'll see almost no applications. It's very abstract. Compare that to my 345 course where we did some very applied, real world things. Combinatorics is used in computer science and computer algorithms, statistics is used to model tons of real world situations. I believe Nancy Mendell works in biostatistics. Everyone in the AMS major can place one of their feet in the theoretical world and but also firmly place the other foot in real world application, and hence industry.