Difference between fourier transform and first fourier transform?

The question almost certainly intends "fast" instead of "first". The difference between a Fourier Transform and a Fast Fourier Transform is only the amount of effort required to generate the result. Both have the same the result.

The original Fourier Transform requires an amount of effort which is proportional to the square of the amount of data being used. So if the amount of data doubles, the amount of effort to calculate the result quadruples. In contrast, the subsequently discovered Fast Fourier Transform requires an amount of effort proportional to the product of the amount of data and the base-two logarithm of the amount of data.

Thus, if the amount of data doubles, the amount of effort increases but by less than a quadruple. With each doubling of the data size, the amount of effort increases by a diminishing factor which slowly drops toward but never reaches two.

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