The purpose of arranging all diagonal elements in descending order is to allow splitting R into two parts if A is low-rank or close to it. Here's an example:
>> A = [0 3 -2 1; 0 -6 4 -2; 0 2 -3 -1; 0 1 1 2];
>> [Q, R] = qr(A); R
R =
0 3.0000 -2.0000 1.0000
0 6.4031 -4.5290 1.8741
0 0 2.3426 2.3426
0 0 0 -0.0000
>> [Q, R, P] = qr(A); R
R =
-7.0711 -2.1213 4.9497 0
0 2.3452 2.3452 0
0 0 -0.0000 0
0 0 0 0
In the second case, it's easy to see from the diagonal of R that A is of rank 2 (accounting for some round-off error). Knowing this, One can use A*P = Q(:, 1:2)*R(1:2, :) as a decomposition of A that has this low rank. This would be much harder to do using Q and R from the two-output syntax of the qr function.
Here are some additional references: https://en.wikipedia.org/wiki/QR_decomposition#Column_pivoting and A BLAS-3 Version of the QR Factorization with Column Pivoting (this paper discusses details of efficiently implementing QR, but the introduction should provide more context about where column pivoting is used).
