It's important to remember that in almost every way that matters, the two approaches are equivalent. But since the various political parties are all trying to squeeze out as much torque as possible from their own proposals, this point is easy to overlook. So as a public service, here is the Econ 101 explanation of how the two policies work, and why they are equivalent.
Before the policy, the intersection of the supply and demand curves for ghg-emitting products - point A on the graphs - will generate emissions equal to Q0, and the price will be P0. Suppose that the government wants to reduce the quantity to Q1.
- Carbon tax: Suppose that a carbon tax π is added into the price. For a given quantity, the supplier's price will be the old price plus the amount of the tax, and the supply curve will shift up to S*. The new equilibrium is at point B, the quantity is the target Q1, and the price will increase to P1. Note that the price increase will be less than the tax, although if the demand curve is fairly steep (i.e., inelastic, or relatively insensitive to changes in price), the increase in the price will be pretty close to π.
- Cap-and-trade: Suppose that the government restricts emissions to a level consistent with Q1. The new supply curve - denoted by S* - is now vertical at the target: no matter how high the price goes, supply will remain fixed at Q1. The new equilibrium is again B: the quantity is determined by the cap at Q1, and the price will rise to P1.
What distinguishes the two is what happens to π - the difference between the price the consumers pay at B and what it costs suppliers to produce at Q1. In the case of the carbon tax, the money goes to the government. But if output is capped at Q1, that difference is pure profit: a permit to produce one unit of output allows its owner to collect a rent equal to to the difference between the selling price and the cost of production. If permits are traded, their price will be bid up so that their price will be equal to π. So where that money goes depends on how the permits are allocated in the first place. If the permits are simply given to existing emitters, then those profits are pocketed by the firms. If the permits are auctioned off, the price will be bid up to π, and the government gets the money.
So if permits are auctioned off by the government, then cap-and-trade and a carbon tax are equivalent: same quantities, same prices, and the government gets revenues equal to the area in the green rectangle in the graphs.
Update: Mark Thoma reminds us of a more sophisticated - yet still very accessible - analysis at Environmental Economics.