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Absolute colorimetry and relative colorimetry actually use the same table but differ in the adjustment for the white point media. If the output device has a much larger gamut than the source profile, i.e., all the colors in the source can be represented in the output, using the absolute colorimetry rendering intent would "ideally" (ignoring noise, precision, etc.) give an exact output of the specified CIELAB values. Perceptually, the colors may appear incorrect, but instrument measurements of the resulting output would match the source. Colors outside of the proof print system's possible color are mapped to the boundary of the color gamut. Absolute colorimetry is useful to get an exact specified color (e.g., IBM blue), or to quantify the accuracy of mapping
The goal in relative colorimetry is to be truthful to the specified color, with only a correction for the media. Relative colorimetry is useful in proofing applications, since you are using it to get an idea of how a print on one device will appear on a different device. Media differences are the only thing you really would like to adjust for. Obviously there has to be some gamut mapping going on also. Usually this is done in a way where hue and lightness are maintained at the cost of reduced saturation. Relative colorimetric is the default rendering intent on most systems.
The perceptual and saturation intents are where the results really depend upon the profile maker. This is even how some of the competitors in this market differentiate themselves. These intents should be created by the profile maker so that pleasing images occur with the perceptual intent while eye-catching business graphics occur with the saturation intent. This is achieved through the use of different perceptual remaps of the data as well as different gamut mapping methods. Perceptual rendering is recommended for color separation.
In practice, photographers almost always use relative or perceptual intent, as for natural images, absolute causes color cast, while saturation produces unnatural colors. Relative intent handles out-of-gamut by clipping (burning) these colors to the edge of the gamut, leaving in-gamut colors unchanged, while perceptual intent smoothly moves out-of-gamut colors into gamut, preserving gradations, but distorts in-gamut colors in the process. If an entire image is in-gamut, relative is perfect, but when there are out of gamut colors, which is more preferable depends on a case-by-case basis.
Saturation intent is most useful in charts and diagrams, where there is a discrete palette of colors that the designer wants saturated to make them intense, but where specific hue is less important.