essentially, in computing the naive DFT directly from the summation:

$$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \, e^{j 2 \pi \frac{nk}{N}} $$

there are $N$ table lookups for the twiddle factor $ e^{j 2 \pi \frac{nk}{N}} $, $N$ complex multiplications, and $N-1$ additions. and that's just for one value of $X[k]$ and one instance of $k$. then the naive DFT throws away all of that intermediate data away and goes through all of it again for $X[k+1]$.

1. so the FFT holds on to some intermediate data.
2. the FFT will also make use of factoring the twiddle factor a bit so that the same factor can be used for an intermediate combination of data.

essentially, in computing the naive DFT directly from the summation:

$$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \, e^{-j 2 \pi \frac{nk}{N}} $$

there are $N$ table lookups for the twiddle factor $ e^{-j 2 \pi \frac{nk}{N}} $, $N$ complex multiplications, and $N-1$ additions. and that's just for one value of $X[k]$ and one instance of $k$. then the naive DFT throws away all of that intermediate data away and goes through all of it again for $X[k+1]$.

1. so the FFT holds on to some intermediate data.
2. the FFT will also make use of factoring the twiddle factor a bit so that the same factor can be used for an intermediate combination of data.