In some situations, like when you have vast expanses of flat color, it makes sense to stick to web-safe values.
Similarly, if you have a GIF logo and you don't want the main parts of the type to go speckled on 10% of the monitors out there, then make sure that you choose a web-safe color for the body and background of the type.
Don't try to force the whole GIF to web-safe. Anti-aliasing type has the effect of generating about 20 intermediate hues between the foreground and background colors, you can safely reduce the number of intermediate colors by saving the GIF at a lower bit-depth. 4-Bits (16 colors) works well for any one colour of type on any single background color. If you have more than one type or background color, you will need to increase the bit-depth to 5 or 6.
Using the web-safe Palette will eventually frustrate all designers at some point - the lack of pastel tints, too few neutral greys, no exact match for required Pantone colors. There are techniques to improve on these restrictions, but only go part of the way to solving the problems.
Color expert, Bob Stein of Visibone International Color Laboratories, has this to say about the limitations of the web-safe Palette.
The web-safe palette is uniform numerically but it leaves a lot to be desired from the point of view behind human eyes. To me this reveals something about the viewer. Human eyes (and brains) were designed to distinguish very fine shades of brown, the colour of soil, rotten food and faces. Notice that more than half the palette could be called greens and blues. Vast swaths have little distinction in human perception. But blink and you'll miss the earth tones, coarsely glossed over in the leap from red to yellow.
In a similar vein, the eye can discriminate some very subtle shades among the pale pastels, the colours of unripe fruit, the harvest moon and a young girl's blush. These are also given broad irreverent strokes in the web-safe palette.
A contact sheet of photos I took along the Peace River in Florida suggested to me that our ancestral environment was painted almost exclusively in greens, blues and browns. I'm not sure I can explain why the greens aren't more finely classified by the eye-brain machinery. One would think they have a lot to do with the health of a plant and the edibility of its foods. On the other hand, the majority of what's green has no nutritive value.
So if we were to redesign the web-safe palette today, might we not drag and drop the center of gravity significantly toward the browner and the paler? We would honor the woods and the fleshes, the caramels and salmons, the sands and siennas. We would not take for granted the lavenders and limes, the vanillas and seafoams, the pumpkin seeds and cucumber cores, the brutal arctic skies and the shy blossoming roses.
It is obvious that the web-safe Palette was invented by a scientist and not an artist, it is a compromise, and at this stage in the development of internet-based media, it is not going to change.