Extra payment on a loan: shorten the term or lower the instalment

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This assumes the fixed instalment arrangement, where every payment is the same and the split between interest and principal moves over time. Everything is worked out in closed form rather than month by month, so a long term does not accumulate rounding.

Results

The last row is the one that settles the argument, and it is computed rather than asserted. Bring both futures back to the day of the extra payment at the contract's own rate and each of them is worth exactly what you paid in. Not roughly, not nearly: to the cent, for any amount and any month.

So the enormous gap in the row above it is calendar and not money. Shortening the term stops payments that were going to happen far away, and money far away is worth less; lowering the instalment gives a little back every month starting now. The totals differ because they are added up at different distances, not because one of them found value the other missed.

The same extra payment, made at different times

Read the two columns against each other. The middle one falls away as the years pass, from a headline figure early on to a modest one late, and the right hand one does not move at all. Every row is the same amount of money doing the same job; only the number of future payments it cancels changes.

That does not make paying early pointless. It makes the reason for it something other than the saving: an earlier payment removes more months of owing, and owing has consequences that no discount rate captures, from what happens if income stops to what else the debt prevents you from doing.

There is one number here that is a rate rather than a total, and it is the one worth carrying away: an extra payment earns exactly the interest rate of the contract, with no tax and no uncertainty. Whether to make it at all is a comparison against what the same money would earn elsewhere, after tax, and never against the headline saving, which is not a rate at all.

Which of the two to choose is then a question about your month rather than your arithmetic. Lowering the instalment buys room in the budget now; shortening the term buys an earlier end. They cost the same, so pick the one whose risk you would rather carry.