Get DVD-quality video.
Many PlayStation 2 and virtually all Xbox games support 480p progressive-scan resolution--some Xbox games can even output 720p and 1080i images. The Xbox 360 is the current king of HD, as its games play in 720p and 1080i. Playing games at 480p, which is supported on all HDTVs, yields a DVD-equivalent image that's twice as sharp as what you'd get at standard resolution. And in order to get progressive-scan, you'll need to connect via component video; composite and S-Video connections aren't compatible.
Wide-screen means more gaming real estate.
All three previous generation consoles--Xbox, PS2, and GameCube--support native wide-screen (16:9) resolution in at least some of their games. The Xbox 360 utlilized widescreen in all of them. That means a more panoramic view of the battlefield, court, gridiron, ballpark, maze, or heavens, depending on your genre of choice. And games that support wide-screen will be properly proportioned, not artificially stretched or bordered by black pillars on the sides.
It's well worth the $20.
Any tech enthusiast's natural inclination is to use as much of their gear to the best of its ability as possible. Maximizing your gaming experience is a no-brainer and doesn't cost too much extra: just $20 or so for the component-video adapters.