One half of the problem is light. Light is an oscillating electromagnetic field. The frequency of oscillation determines the color. Higher frequencies have shorter wavelengths and are blue. Lower frequencies have longer wavelengths are red.
Light is a mix of a range of wavelengths. Sunlight contains a wide range of wavelengths, including some that are too long or short for us to see. A laser contains a very narrow range.
How light interacts with matter is the other half of the problem. Electromagnetic fields exert forces on electrons. Electrons in atoms or molecules are constrained in how they can move.
Metals have electrons that can move freely. This makes them good conductors. Freely moving electrons reflect light. Light makes electrons vibrate and absorb the light. Vibrating free electrons emit light and stop vibrating. Thus metals are shiny.
But even in metals, electrons are constrained. For example in copper, electrons can vibrate more readily at low frequencies. Copper reflects red light better than blue. It reflects better still in the infrared and even worse in the ultraviolet. Here is a graph from a random website, http://www.minoanatlantis.com/Minoan_Mirror_Web.php
![enter image description here](https://i.stack.imgur.com/0tnUP.jpg)
So if you shine a red laser on copper, it reflects almost all of it. If you shine a green laser on copper, it would reflect less.
Other materials have different ways of interacting with light. But at each wavelength, they may reflect some and absorb some. They reflect more at some wavelengths and less at others. Even black materials reflect some light.
The answer to this question depends on various aspects. For instance when you say white object, do you mean perfectly white? or white with respect to only visible light? Same happens for a mirror too.
The most direct way to look into the problem would be following. Mirrors are nothing but extremely fine and optically flat white surfaces at the back of glass. Usually its silver coating. Silver when left alone as a lump can act as a white object too. Now we can reformulate the question by asking: Will a fine surface of silver absorb less heat than a lump of silver? The answer is their absorption co-efficient is same. So heat absorbed per unit mass will be same.
But mirror looks more brighter as if it reflects more light than white object. The answer to this is that usually white objects do not have flat surface. So the reflection is not regular unlike mirror.
P.S. If you consider the glass of mirror through which light has to pass, you'll find that some light is absorbed their. So reflectivity of mirror is actually less.
Best Answer
The difference is the direction the light is emitted in. Mirrors 'bounce' light in a predictable direction, white objects scatter light.