Do note @tswayne’s comment that you won’t get an absolute measure of the number of molecules, only an approximate concentration relative to other cells in the section, and compared to other sections if you have stained in exactly the same way (and preferably at the same time).
Also, if you follow the link and read the example about dot blots, be aware that the example is a bit old, and only shows the IntDen, as ImageJ used to give. ImageJ now gives IntDen and RawIntDen. Only RawIntDen is actually the integrated density (ie, sum of all pixels in the ROI), and IntDen is the integrated density relative to the scaling of the image. So if the image is unscaled (as you can see at the top of the image title bar it will only say n x y pixels), the two values are the same, but if it is scaled (it will say p x q microns (or whatever) then give the size in pixels in parentheses), then the IntDen is the mean intensity per scaling, eg mean per micron squared if it is in microns. This is essentially the same as taking the mean, except that is mean per pixel.
So to go back to your question about when Integrated Density is preferred, it is useful if you wish to know the absolute intensity per cell for example, but irrespective of cell size, in which case the RawIntDen value can be used. In practice, this is trickier for quantification. For example, have you imaged the entire cell volume in every measured cell?