White color means very little optical absorption, so why a white flat surface (such as a wall) appears as white and not as a mirror-like surface?
A mirror or (almost) any other well polished material, exhibit specular reflection, in which the law of reflection is observed (angle of incidence equals angle of reflection).
Virtually all materials can give specular reflection, provided that their surface can be polished to eliminate irregularities comparable with light wavelength (a fraction of a micrometer). (Wikipedia)
A white wall has surface irregularities larger than visible light wavelength, such that reflection occurs in all directions so no image is formed. But still a white wall appears as white because it saturates the eye with all the wavelengths reflected from it.
Most white objects we see, are transparent, with inclusions. Milk, for example, has small lipid globules in an aqueous suspension. Clouds have ice crystals or water droplets (or both) in air. Pale woods, and paper, have open tubes of transparent cellulose, with lignin between tubes and voids in the tube centers. The white appearance is due to multiple kinds of scattering, and multiple small-angle scattering of incident light is the usual path for incident light to 'reflect' off such an object. So, any image one might see in the uniform reflection of a smooth surface, is scattered and confused by the white object's internal structure.
You can, barely, see a reflection in a milk surface (against a background of white). More commonly, you can see 'local color', where the color of nearby objects is visible in the white object nearby. Artists have to know this, to make realistic images, but we rarely note the effect consciously.
White flat surfaces are white and not mirror like reflective because of the irregularities of the surface that that causes diffused reflection. Due to this diffused reflection the light rays that are coming to the surface gets reflected in an irregular manner and thus do not shine as they do in case of mirrors. Diffused or irregular reflection occurs when light rays coming in one direction strike any uneven surface and gets reflected in various directions. Thus, preventing the surface to shine. In case of mirrors, the surface is very smooth and the light rays incident gets reflected at an angle equal to the angle of incidence and since all light rays come parallel at the same direction of our vision. The surface seems to shine.