Why do people insist on comparing only things that are same? The same things are same, no point whatsoever comparing them: they're same! No, you should only compare things that are different, that way you learn more about their differences.
And what's better, an umbrella or a toaster, depends on the one's goals: if one'd like to fry some bread, toaster is more useful; if one'd like to cover from rain, umbrella is far superior, although toaster could used be too; if one'd like to drive a nail into the wall, toaster again is better; etc.
My statement was as approximate as the one I was replying to. If the original statement had been "you can use FFT to implement some operations involving matrix multiplication faster" I wouldn't have taken issue with it. But saying "FFT is faster than matrix multiplication" has little meaning.
The naive implementation of the DFT in terms of a matrix-vector multiplication has time complexity O(n^2) [1]. The point is that the FFT approach is much faster than that implementation.
And what's better, an umbrella or a toaster, depends on the one's goals: if one'd like to fry some bread, toaster is more useful; if one'd like to cover from rain, umbrella is far superior, although toaster could used be too; if one'd like to drive a nail into the wall, toaster again is better; etc.