It may be useful to recognize that mathematical symbols represent things, whereas computer programs provide instructions. The sigma thingy represents the sum of some products; for-loop tells the computer how to compute the sum of those products.

Thus, we may be able to do some algebra with the sigma version that never requires us actually to compute its value. We can see, for example, that the sigma version with the 4 at the top minus the same thing with a 3 on top equals 12, without having to calculate either sum. A computer can't do that. A computer does the two for-loops and subtracts one from the other.