What a meter really is
A meter prices time and distance as they happen. If the traffic is terrible, the number climbs. If a road is closed, the number climbs. You carry that risk, and you don't find out what it cost until you have arrived and you can't do anything about it.
What a fixed fare really is
The driver carries the risk instead. He looks at the route, prices it, and eats the difference if it goes badly. That is why it is a slightly higher number on a perfect day, and a dramatically lower one on a bad day, at night, or on a holiday.
When the meter genuinely wins
Short hops in light traffic. A two-kilometre ride across the city centre at 2pm on a Tuesday will be cheaper on a meter than on any flat rate. We won't pretend otherwise.
For anything longer than that — and for anything at all after 20:30, when the night tariff engages — the fixed fare is not close.
The part that isn't about money
You know the number before you get in. On a 4am airport run with a flight to catch, that is worth something on its own.