Why do the fibre wads in some cartridges disintegrate into lots of little pieces, and others don?t?

Shotgun cartridges.
Do you mean that ‘falling confetti’ effect you sometimes get when you fire a fibre-wadded cartridge?

If so, then I once got curious, so fired one into a straw bale at short range, and dug out the wad.

A layer (or maybe several layers) of thin, shiny paper on the back of the wad had been burnt off, but otherwise the wad was intact.

If a wad does completely break up outside the barrel, so what?

It has already done its job as a gas seal between the powder and the shot and ? arguably ? if it does break up, then it should degrade more quickly.