Economically, the difference is irrelevant. Let's say you need to boil one liter of water. The specific heat of water is about 4.2 joule per gram degree Celsius. Meaning, for every gram of water that we want to make hotter by one °C, we must supply one joule of energy. A liter is 1000 grams, and let's say the cold water starts at 15°C, and we want to go to boiling, 100°C.
So, we need to increase the temperature of 1000g of water by 85°C. The energy required is: 1000g * 85°C * 4.2 J/(g °C) = 357000J = 357kJ
Now, my last natural gas bill I used 1900 cubic feet gas for $24.56 (USD). That's 1.29 cents per cubic foot. One cubic foot contains about 1000kJ of energy. Thus, natural gas energy costs 0.00129 cents per kJ.
Next, we must account for efficiency. 90% or better of the energy of gas is available as heat with simple combustion, but if you put your hand around a stove it's easy to feel that a lot of that heat is not going into the water. Actual efficiency is going to depend a lot on your stove and your cookware, but let's just say the efficiency is 5%. That raises our energy requirement to 357kJ / 0.05 = 7140kJ.
We can then calculate the total cost: 7140kJ * 0.00129 = 9.2 cents. This is such a tiny cost that any discussion of savings is purely academic.
Even so, using hot water from the tap won't save very much. Let's just say you have a magic water heater that makes hot water for free. It isn't boiling, though: an oft-recommended temperature is 120°F or 49°C. Taking our previous assumption of cold water being at 15°C, that means the water heater has raised the temperature by 34°C, leaving the stove to raise it another 51°C. So, given a magical water heater, at best your costs can be reduced by 40%. Of course, this doesn't take into account:
- All the water you run waiting for the tap to become hot is wasted hot water, which just sits in the pipe when you are done.
- Water heaters are not magic. (typical efficiency is between 70% and 98%)
- In addition to the energy required to heat the water to boiling, it takes quite a lot of energy to get the water to continue boiling, which must be done at the stove.
All of these things eat away at the efficiency improvement of using the water heater. If your heater is not especially efficient and the pipe to the faucet is long, it might be less efficient to start with hot water.
If your goal is efficiency, best would be to improve upon the 5% efficiency of water heating at the stove. This can be done by adding a heat exchanger to the pot, which is common on camping cookware:
I have also seen, in one of those expensive cooking stores in the mall, a pot with heat exchanger of more traditional kitchen-weight construction.