Being someone who cherishes the ability to do complex maths without a calculator, and having friends who rely on Wolfram, I think that the difference is that when you know the mathematics, you see the mathematics. What you see as a trivial problem that is simply a solution to a homogenous second order linear differential equation, your friend who doesn't know the mathematics sees as an entirely different problem. I'm not saying that you can get by indefinitely without mathematics, but that a lot of problems that have mathematically elegant solutions have other (probably less elegant) solutions, too.

I'm not even talking very advanced stuff. I wouldn't say my math is particularly good. I've forgotten most of the specifics of what was taught in those classes. I totally use Wolfram Alpha when I can, I don't do math for the thrill of doing it, but because I have a problem and I'm trying to find a solution. I think it's more important to have the ability to break down a problem mathematically, and then read up on the parts you need than to remember everything by rote. The problem is that if you've never taken any advanced math, it's very difficult to read up on it and knowing where to start.

I'm not a programmer but. . .unless you're coding an explicitly math-intensive application, aren't there tools that could do any math that's required? Of course, the tool would have to come from a trusted source.

If you don't know the math, how do you see what math applies?

Like I said, might not be a problem for a simple web app, but have you ever tried writing a game? For even the simplest game I end up using a lot of (fairly simple) math, linear algebra and some physics. If you start doing stuff in 3D, you'll need even more.