Brad:
You are treating an FFT as if it were a spectrum analyzer which produces a magnitude or energy profile of how much signal is at a particular frequency. The FFT does much more than that. It tells not only what magnitude is at a frequency but what phase angle the signal has there.
Let's take an example:
You would not want the fft of sin(t)+cos(2t) to be the same as sin(t)+sin(2t). You would want the result to show that the stuff at 2t is 90 degrees out of phase depending on what signal you input.
Bob