In (much) older compilers, I believe they would break (boolean == false) into 2 register assignments and a compare code in machine language. The first example would be broken into one assignment and a NOT operator. In terms of performance, the compare operation would take a number of clock cycles, depending on the size of the register being compared, compared to a bitwise invert (1 clock) and would be slower to execute.

That being said, I believe newer compilers do away with this, so it should be OK to go with either.