Reasons not to use web safe colors:
1) Deprecation
Microsoft stopped supporting Windows 3.1 and IE6 due to the advancement in technology plus it's too expensive and time consuming to dedicate resources to manage and service such older platforms. Modern displays are now 24-bit true color. Father time in the digital world dictates what technology to use or not use.
2) Limitations
X11 colors (browser created) are not synched with HTML 4.01+ standard colors. 256 color displays still dither (i.e. the user who commented about the Google site not looking right). Web safe vs. "really safe" 216 colors is simply confusing and really only 22 colors render as expected on older displays anyway, the whole concept is absurd.
3) Accessibility concerns
For screen readers, web safe colors do not make things any easier as some of those kinds of browsers do not use colors. Sticking to web safe for accessibility concerns is a common misconception. Follow the same rules for contrast and foreground/background no matter which palette used.
4) HSL and the CSS3 spec is coming
The future looks promising, the constantly evolving CSS3 spec deprecates system colors (X11) and will likely use the HSL (Hue, saturation and lightness) color system which was actually developed in the 70's but and supports high quality resolution and capabilities of contemporary graphics chipsets and browser support. Commands via CSS3 such as this:
Note: RGB is still in the spec and alpha channel values exist for both for those who know about transparency and other filter effects.