It's a bit complicated to explain, but in general it does not increase the speed / power by 2x. It depends on what you do and how well the program was designed to make use of 4 cores.
The vast majority of games only use 2 cores so having 4 cores does not improve performance. There are very few games that can use 4 cores, but not effectively. For example, I saw a CPU benchmark last year for a game that can actually use 4 cores (I believe it was Mafia II).
Basically the 3rd and 4th cores were de-activated to see how well the game performed. The 3rd core was then activated to see the improvement in performance. Finally the 4th core was activate to measure the increase in performance. When the 3rd core was activated, performance increased by about 25% which is pretty decent. However, activating the 4th core only improved performance by about another 6%.
To reiterate, performance depends on how well the program has been designed to use more than 2 cores; if at all.