A colorimetric test is a test which forms a color. The amount of the color is then measured. In most tests the more color formed, the more of the test substance there is in the water. You could look at the color with your eyes and compare the color formed with some standards. This is a visual measurement of the color.
Visual methods can give very good results when applied in an appropriate manner.
However, visual measurements of color are subjective. Everyone's eyes are different. Some people are color blind. Colors might look different when there is different lighting. When your testing protocol requires the utmost accuracy and precision these problems must be solved.