When dealing with Fourier analysis, you need to be careful with terminology. The fast Fourier transform (FFT) is an efficient implementation of the discrete Fourier Transform (DFT). There is also the discrete-time Fourier transform (DTFT) which under some stimulus conditions is identical to the DFT. There are also continuous time Fourier transforms.
You seem to be comparing a continuous-time Fourier transform (your integral) with a discrete-time Fourier transform. It is not surprising to me that you are getting different results.
Steve Eddins addressed these issues nicely in his blog a few years ago: (see blog posts starting on Novermber 23 2009):
