Acidification before an acid or neutral fix is said to reduce the potential for fixer stain and formation of dichroic fog in some fixers.
If you use an acid hardener fix, you can precipitate the alum out of the fixer onto your film or paper by going from developer to water to fix. The alum is difficult if not impossible to remove and can be seen as a white crust or sheen on negatives or prints. So, a stop bath is really a good idea with acid hardening fixes.
It has also been shown that uniformity in negatives and prints is somewhat better when a stop bath is used after the developer. The best example illustrating that is with the RA paper example above. Uniformity is best with a stop after development and before the blix. On large prints and LF or MF negatives, you can often see nonuniformities, streaks or mottle in large areas of uniform density.
PE