TL;DR: Whiskey stones work by absorbing heat from the whiskey in an attempt to reach thermal equilibrium1.
As Thomas mentioned, ice has three cooling effects:
- Ice itself takes 2.11 kilojoules of heat per g to have its temperature increased by 1 degree (Celsius). This number is known as "specific heat capacity"
- Ice takes 334 kilojoules of heat per kg at 0 Celsius to melt. This number is "Latent heat of fusion/melting". Note that this is a lot compared to the rest; heating ice from -15 degrees to 0 will require a tenth of the energy that you will need to melt it afterwards.
- Water takes 4.18 kilojoules of heat per kg to have its temperature increased by a degree.
Note that specific heat works both ways; the same number applies for an increase of temperature/inflow of heat and a decrease of temperature/loss of heat
When you chill a whiskey the normal way, ice absorbs heat in different ways, cooling the whiskey (which itself has a specific heat capacity of 3.4 kJ/kgK) till the system is at the same temperature. Usually, what happens is this
2: The whole of the ice reaches 0 Celsius, and the whiskey is significantly cooled. Once this happens, the ice melts and drops the temperature of whiskey significantly. Finally, the ice(now water) heats up a bit.
Now, looking at soapstone (what they use in whiskey stones): Soapstone is thrice as dense as ice, but only half as effective in absorbing heat (specific heat capacity 0.9). So it will be a 50% more effective than the same number of ice cubes during the "reach 0 degrees" phase. After this, ice wins by a huge margin -- the same back-of-the-envelope calculation (where I take the same volume of soapstone, giving a 3:10 ratio for mass) gives me 28C as the final temperature. Of course, this can easily be remedied by using more whiskey rocks (I think that my estimate for the 1:10 ratio in the ice case may be wrong though -- I don't drink and I don't know how much ice typically goes into a glass of whiskey).
Note that another important factor is thermal conductivity. Ice has a thermal conductivity of 2.18 W/(m·K) and water has 0.56 W/(m·K), while soapstone has 6.4 W/(m·K), approximately thrice as much as ice.
I guess that it takes a while for the ice to reach the 0C state uniformly. Instead, just the outer layer will be at 0C, and there will be a drastic temperature gradient on going inwards. The outer layer will slowly melt (this is a negligible effect since it's a very thin layer that is melting3). On the other hand, soapstone is able to absorb heat at a faster pace (and it is able to absorb more heat than non-melting ice). Which may partially balance out the drastic differences mentioned above -- after all, one doesn't wait for thermal equilibrium to be reached before drinking a whiskey. So the faster chilling from soapstone may level the playing field here -- as long as ice isn't allowed to melt in bulk, it ought to be fine.
As far as chilling effects go, ice cubes are more effective in the long run, but whiskey stones will initially chill faster than ice. And of course, the stones have the additional benefit of not diluting your drink.
1. Thermal equilibrium is when there is no longer any net heat exchange between components of a system. This usually occurs when all the temperatures are equal.
2. Assuming that the mass of ice in a glass of whiskey is a tenth of the mass of whiskey itself, and that ice started off at -15C and whiskey at 30C, we need 3.4*30*10=1020 units of energy to bring whiskey down to 0C. We get 2.11*15=31 units from the warming up of ice, and we have a potential 334 more from the melting. Clearly, this is not enough, so the water formed must also heat up above 0 degrees C. A back-of-the-envelope calculation gives me 17 degrees as the final temperature for this mix.
3. I may verify this later; not in the mood to solve differential equations now :)