Mortgage payoff statement vs just making very large additional principal payment

None, really. You can try to guess how much interest has accumulated since your last statement date and include that in your "final" payment, then wait to see if you over- or underpaid. The purpose of requesting a payoff is to have your lender compute the accumulated interest for you (based on your estimated payment date), with a fee for the convenience, so that you don't have to make a subsequent "final final" payment.

I have done that just recently (submitted a ‘nearly payoff’ extra payment), and it worked perfectly fine. I intentionally paid about 500$ low, and let them make a final deduction at the end of the month.

If you don’t want to wait till end of month, overpay by a 100 $ or so to be on the sure side. They’ll mail you a check back for the difference.

If you request a payoff statement, one disadvantage is that it will include the full interest for the validity period (often several weeks), and you won’t get that back - you basically pay them an extra some weeks interest.