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.