Percentage change has a direction — it needs a "before"
Percentage change = (New − Old) ÷ |Old| × 100. It only makes sense when one value is genuinely the starting point and the other is what it became — a price before and after a markup, a population last year versus this year, revenue in one quarter versus the next. Because it divides by the OLD value specifically, going up and coming back down are not mirror images: a 60% increase from 50 to 80 is undone by a 37.5% decrease from 80 back to 50, not another 60% decrease.
Percentage difference has no direction — it needs no "before"
Percentage difference = |A − B| ÷ ((|A| + |B|) ÷ 2) × 100. It treats both values as equal, simultaneous measurements with no starting point — like comparing two competing products' prices, or two labs' independent measurements of the same quantity. Because it divides by the AVERAGE of both values rather than either one specifically, swapping which value you call "A" and which you call "B" doesn't change the answer at all — order genuinely doesn't matter here, unlike percentage change.
A worked example showing all three numbers
Take 50 and 80. Percentage change from 50 to 80 is +60% (change of 30, divided by the old value 50). Percentage change from 80 down to 50 is −37.5% (change of −30, divided by the old value 80) — a different magnitude, because the base value changed. Percentage difference between 50 and 80 (in either order) is 46.15% (a gap of 30, divided by the average of 50 and 80, which is 65) — a third number, because it's answering a genuinely different question: not "how much did it move" but "how far apart are these two values, relative to their scale."
Picking the right one for your situation
Use percentage change whenever there's a real before-and-after relationship: prices over time, growth rates, salary changes, year-over-year comparisons. Use percentage difference when comparing two independent values with no temporal or causal order between them: two vendors' quotes, two lab measurements of the same constant, two competitors' market shares at the same point in time. Using percentage change when there's no real "before" silently picks one value as the baseline arbitrarily, which can make a comparison look more one-sided than it actually is.