How useful is a degree in Applied Mathematics for a career in Games Programming compared to other degrees?

First, let me say I'm not in that industry so take my answer with a grain of salt.  My thoughts are that a computer science degree is probably more appropriate for someone who wants to be a game programmer.   Of course, this likely depends on the aspect of the gaming industry you'd like to focus on.  Depending on the type of Applied Mathematics you'd like to do, you will be required to do programming.  However, the training you will receive in programming as an Applied Mathematician is quite a bit different than the training you will get as a programmer in computer science.  One of my computer science buddys is always joking that we applied math guys don't really know how to program.  And from his prospective, he has a point.  By the way, he's a gamer as well.

Mathematics does, however, play a significant role in gaming.  Especially with graphics.  This may be more pure mathematics than applied though.

I have an applied math degree and have since moved on into seismology.  All the problems I have to solve require programming which is a skill you learn merely by doing, and requires nothing more than consistency, a little thought, and a lot of coffee.  I believe my math degree helps me when I work with data smoothing, interpolation, extrapolation ideas, I've never done krigging but I'd imagine that is a useful skill as well.  Also signals processing has some neat tricks for cleaning and representing data, they can also teach you digital compression ideas fairly quickly.  The fundamental concept I had to learn after the degree was that the problems need to be discretized for a computer to handle, but that wasn't a far stretch because I've been exposed to discretizing problems and trying to get them in a form ready to be used on a computer.  At the end of the day, I think with the applied math background you will have the tools to seek a model that describes what you plan to simulate and get enough concepts from a numerical computing / linear algebra class to implement.  Many people also major in computer science and math since the degrees are very similar.  What you will be lacking is knowledge of dirty little tricks you can play with computers to make your code run smoother.