Restricting your palette
Restricting your palette is not a bad idea. Many of the most famous painters restricted their palettes to a few oil paint pigments. This was more of a practicality than an aesthetic judgement.
In earlier times, artists had to grind their own pigments from locally available raw materials. The broad spectrum of paint colors available today is largely based on synthetic, chemically derived pigments that just weren't available until recent times. Some of the traditional paint pigment materials, like cadmium, are now prohibitively expensive.
It is much easier to make color choices if you have less colors to choose from! Bob Stein also points out that almost half of the colours in the web-safe palette are so close to their neighbours that they might as well not be there. So, you only have about 120 distinctly different colors anyway. That's still way to many to juggle with when it comes to creating a color scheme for a Web page.