One should expect a significant improvement
After all, he's going from a tired, well over 500 charge/discharge cycles battery (almost 1000 cycles if he got it near launch) to a brand-new sub-10 cycles battery.*
If nothing else changed, that alone would give you around a 25% increase in battery life - these kinds of cell are generally rated for 80% capacity after 500 cycles.
You'll get the same improvement if you swap the battery in any other smartphone after two or three years of heavy usage.
If it doesn't, then something is quite badly wrong - either they've put a smaller battery in there, the phone is drawing more power, or the battery is faulty.
My guess is that Apple made a mistake in handling multi-core in the low-power environment and either spin both all the time or fluffed concurrency.
The issues on older phones may well be user colouring - someone's mentioned battery life so the user tests and finds that it doesn't last like it did when new, they've put this new firmware on it, ergo it's the new firmware.
Or it could just be an old battery. Easy test - replace the battery. Oh yeah...
(*Assuming he used it instead of, I dunno, putting it in a shrine and praying to it.)