1) Hardware knowledge. I didn't start programming until I started college, but I quickly realized that, growing up as a pretty avid gamer, I still had one advantage that a lot of the CS majors didn't have: I knew more about computers than they did. Games are some of the most demanding applications being created today, and, as a result, I had spent a lot of time reading hardware reviews and benchmarks. This resulted in my having a somewhat innate understanding of how data is passed around inside a computer and how this affects performance. It takes time to understand on a deep level how your code is interacting with the hardware, but when you see the frame rate tank because you used an inefficient algorithm somewhere, it sticks with you instantly.[1]
2) Instant gratification. Much like web programming, game programming allows you to see the result of your work, immediately and incrementally. And it looks good, because you get to use professionally developed resources (models, textures, etc). A side benefit is having good looking programs to put in your portfolio.
3) Wide variety of problem domains. Once you get familiar with programming in general, you can then work on pretty much anything you want: 3D graphics, physics simulation, AI, networking, UI / UX , HCI, etc.
4) It's fun.
Try giving Unity a shot and see how you like it: http://unity3d.com/unity/download/ . It's multiplatform and has API's in C#, JavaScript, and Python, so you have an element of choice. If you're a Windows user, I know a few people who started out by playing around with XNA, which uses C#. Examples of free commercial game SDK's are Epic's Unreal Development Kit (http://www.udk.com/) and Valve's Source Engine SDK, which comes free with any Source Engine-based game.
[1] When working on a graphics project, I accidentally put the function that imports and initializes the texture resources into the draw loop, which meant that new memory was being allocated for them 60+ times per second. Needless to say, I became very aware of the problem when my fans started to spin up like jet engines.