I think you need to balance two parts:
1 - brand recognition of colleges - the source of the degree gets you past the first round of recruiting. Generally, most companies based the list of good schools on the success they've had with previous candidates.
2 - getting the education you want - the door may be open from your school, but if you don't have the knowledge to back it up you probably won't pull in that first job offer. So picking a school with a good web program - if you want to work on web industry products - is just as important. Or else, plan to supplement your education with independant web research.
To answer the questions:
What do other people think is more important based on their experiences?
The skill set you learn. I have trouble justifying a school that doesn't give you the education you want. No matter how awesome the rep of the school is, it's your time and your money and if you don't like the education you'll get there, it's not worth it.
Has anyone here studied in a math-oriented graduate program? If so, what are you working on now?
I have a B.S. in Computational Mathematics, and a M.S. in Computer Science. The CS degree was from a college with the tech school model. I find both useful. The math degree taught me how to look at problems in many different ways. The CS degree taught me how to get stuff done. I'd say I use both, but the direct application of mathematical topics is less obvious.
Is school reputation in the workplace more important than the focus of the program?
Not more important, but possibly equally important.
To those who hire developers - what would you prefer?
This is a tricky question. Hiring practices can vary widely across companies. Working from the enormous, faceless institution model:
The source of the degree and your GPA gets you through HR. Hiring managers may never see your resume without a first pass from HR.
Showing you are smart and get things done should get you through the hiring manager/interview process. This is where your ability to talk about the technical domain will really factor in. A math-based program should not be a detriment here - even if your school didn't cover web topics, self-research and demonstration of hands on knowledge should be enough to get you through.
The thing that wouldn't fly would be coming to a web job interview and saying "I didn't learn it in school, so I don't know..." That's not good, since in a year or two, every technology the school taught could be obsolete, and you'd need to self-educate anyway. Better to be able to say "I chose a math-based program to get a solid understanding of the fundamentals, but then I used technology X, Y, Z in my own personal projects". That's an impressive answer and something that will make managers take note in a good way.
What is the difference in how math and engineering programs are considered in academic circles?
Can't say. I've so completely avoided academic circles.
Have you tried a background check on the hiring profile of candidates? Particularly from school #2? Can they give you a list of companies that have hired recent grads? It might also help to look at social networking sites (like LinkedIn) to see an independant source for what grads are doing now. It might even let you contact alumni to get a sense of what they felt was good or bad about their educations.