If you only have an 8-bit output to your color monitor, this is of course academic, because you'll never be able to see more than 256 colors anyway. These colors are sometimes called process colors, because you use them in four-color process printing. If you have a color printer, you know that the toner in the machine consist of these primary colors. Primary colors mean that all other colors can be created by mixing these colors together. Cyan, magenta and yellow are theoretically all you need, but to make a print look sharp and crisp you also use a black plate in the printing press.
Indexed files contain less data and therefore use less disk space.
If you are designing for the web, you'll have to consider that many computers can't display 24-bit color. There are a lot of people out there who are limited to 256 colors on their monitor. It may well come out looking bad, and it still takes as long to download. There are advantages and disadvantages to both formats, and I'm not going to get into the big discussion about GIF vs JPEG here. The important thing to know is that GIF is indexed and JPEG isn't.