Strassen makes your rounding errors bigger the more levels of recursion you do. We're talking ~7 bits of error due to rounding (in a paper I read).

Real computers are finite precision. There are very real issues with numerical stability where you are running along fine for many many cases, and then there is a catastrophic cancellation that wipes out all of your accuracy. There is a huge difference between writing a paper on bounds of matrix multiplication, and delivering performance in a finite precision world.