I was taught that something which reflects all the colors of light is white. The function of a mirror is the same, it also reflects all light. What's the difference?
Update:
But what if the white object is purely plain and does not scatter light? Will it be a mirror?
The difference is the direction the light is emitted in. Mirrors 'bounce' light in a predictable direction, white objects scatter light.
A white object reflects the light in all the directions, independently of the original direction. It is called a diffuse reflection. If you shine a beam of light onto a white surface, it is scattered in all the directions.
On the other hand, a mirror reflect the light symmetrically to the input direction, with no scattering. This is what allows us to see "through" the mirror.
Edit to add: "Normal" coloured object all have a diffuse reflection, their colour being given by the mix of wavelength they reflect. To better understand the difference between the two kind of reflections, you can also read the related question : "Why can you have shiny black objects?" (and the answers !)
Other answers are correct except that they forget to mention why white objects behave differently than mirrors. Well, generic white objects have very rough bumpy surfaces. This causes scattering.
On the other hand, mirror has a very even and polished surface so that the simple law for specular reflection of incoming rays holds.
Of course, you can also obtain any kind of roughness in between. Actually, you can produce a mirror by starting with a generic white object and polishing it.
White color is associated with reflected (not absorbed) light. White paint usually includes a titanium oxide component, whose absorption is in UV.
The difference between a shiny surface and a Lambertian surface is its roughness, if light is reflected in a collective manner it look shiny. Any smooth surface is shiny given a grazing angle.
A mirror is usually coated with a metal surface, silver for example. Silver's metallic behavior makes most of the light reflected back (about %95 percent reflection, which is why it has a greyish tint), however it becomes transparent for UV light.
A mirror can be engineered to have more reflectance than silver by using multi-layer reflection and interference, using dielectric materials instead of metal. They are called dielectric mirrors.
To sum up, white color is due to non absorbed white light, reflection is due to non transmitted light.
I know it's been a long time, but I wasn't completely satisfied with the answer so I had to experiment myself. A really easy, cheap experiment done on my desk:
I took 2 white pages and my black-surface pencil, laid the pencil on one white paper and blend (to certain point) the other at its side.
I saw perfectly how at certain point, really really close to the pencil, the blended paper turned darker. It couldn't be a shadow from the point the light came from (perpendicular to this point). I was amazed, so I decided to go a bit further.
I took the pencil out and saw how this small reflection disappeared, turning the paper white again, like the paper beneath. Furthermore, just in that small part the paper looked like polished.
To complicate matters, a reasonably smooth flat surface, white or otherwise, becomes a mirror at grazing incidence (the angle of incidence, and the angle of reflection, are close to 90$^\circ$ from the normal direction). This effect often finds application in focusing x-rays, but it works at visible and other wavelengths too. You can see this by putting your eyelevel just above a desk or table, and looking for the reflection of an object placed a cm above the surface at a distance of a few tens of cm away.
I think that maybe its not really on how smooth the surface is because white paint can be as smooth and yet not act like a mirror. Then again snow acts like shattered glass where broken individual snowflakes reflect sunlight making snow look white. It's mainly in the absorption spectrum or the molecular/bonding structure that determines the color of an object, and maybe how the object scatters light, which the electron levels either absorb, reflect, transparent, and/or refract a little bit, some, most, and/or all light. I do think that maybe white objects reflect and scatter white light in all and random directions while mirrors reflect all light in an evenly straight direction.
But I do notice this. Let's say white light coming from the original light source, such as the sun or a light bulb, comes into a plant. The chlorophyll in the leaves will absorb most of the colors in the light and will reflect and scatter mainly green light. When the green light reaches our eyes we see the plant that it is green. If the plant is next to a mirror then the mirror will only reflect the green light from the plant in an evenly straight direction and once it reaches our eyes then we see another image of the green plant in the mirror. If the plant is next to a white object such as a white wall then the white object may absorb the green light reflecting out of the plant, unlike a mirror, but the white object will also reflect and scatter white light from the original light source. So we see the wall as white but we don't see another image of the green plant in the wall. Maybe there are other ways that a mirror is different from a white object but I hope I answered the question.
