Usually, the advice deals with the difference between a 15 year mortgage and a 30 year one. The idea is that the longer you owe, the less the actual value of the money you repay with will be, especially in these inflationary times.
And, in the meantime, the money you don't give to the lender you could be putting into some CD or whatever and earning more than you are paying in interest on your mortgage.
However, that's not to compare paying off one's mortgage entirely with having payments to make for years to come. To that, the same people are apt to say that paying off the mortgage doesn't help your credit rating, while showing your dependability at repaying borrowed money could be important to you if you want to borrow more money.
I think I see a few holes in both of those arguments, but that's what I believe the pros are saying.