BTU/pound of fuel is pretty constant for propane, butane, and white gas (with a slight advantage, 5%-ish, to the lighter compounds).
400-gram butane canister: about $7-8
220-gram butane canister: about $6
gallon of white gas: $10-12 (?it's been a while) at 6+ pounds = 2700+ grams.
So butane canisters, per BTU, are 5 to 8 times more expensive than white gas.
You have to grant canister stoves less priming time, and typically people turn them on and off more because they are so easy to relight, whereas many of us are sorely tempting to keep a white gas stove running so as to avoid having to reprime it.
Putting everything together, I'd say butane costs 4-5 times more. But it is cleaner, easier (in warm weather) and takes less practice to use safely.
Propane in the one-pound cylinders (one pound of steel and 1.025 pounds of propane) is in between. $3.50 for 465 grams of fuel. Over twice the cost of white gas (but more convenient), half the price of butane and good down to almost any temperature (I've used it, with a few tricks, at -41F). But, my, there's a lot of steel to carry in AND OUT – more than with any other fuel. But it's great for car camping.
If I was running a base camp, or melting snow for many people, I'd go with white gas. If I could have planes or horses bring in supplies, I'd go with bulk propane cylinders (my friend with a charter service moves more 20-pound propane cylinders than anything else). If I was far into the third world, I'd get a multi-fuel stove that can take unleaded gasoline, diesel and yak's milk as fuel types – those are even cheaper per liter boiled.
If you're worried about fuel cost or usage, get a pot with a heat-exchanger. That saves fuel and the water boils faster.
Fuel cost is really very small compared to taking time off work, driving to the trailhead, and the occasional hospital bill or gear replacement.