The simple answer is that unlike RGB, HSV separates luma, or the image intensity, from chroma or the color information. This is very useful in many applications. For example, if you want to do histogram equalization of a color image, you probably want to do that only on the intensity component, and leave the color components alone. Otherwise you will get very strange colors.
In computer vision you often want to separate color components from intensity for various reasons, such as robustness to lighting changes, or removing shadows.
Note, however, that HSV is one of many color spaces that separate color from intensity (See YCbCr, Lab, etc.). HSV is often used simply because the code for converting between RGB and HSV is widely available and can also be easily implemented. For example, the Image Processing Toolbox for MATLAB includes functions rgb2hsv and hsv2rgb.